On the explicit representation of the trace space \(H^{\frac{3}{2}}\) and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

Abstract

We consider the problem of describing the traces of functions in \(H^2(\Omega )\) on the boundary of a Lipschitz domain \(\Omega \) of \(\mathbb R^N\), \(N\ge 2\). We provide a definition of those spaces, in particular of \(H^{\frac{3}{2}}(\partial \Omega )\), by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space \(H^1(\Omega )\), based on the classical second order Steklov problem.

Introduction

We consider the trace spaces of functions in \(H^2(\Omega )\) when \(\Omega \) is a bounded Lipschitz domain in \(\mathbb R^N\), briefly \(\Omega \) is of class \(C^{0,1}\), for \(N\ge 2\). It is well known that there exists a linear and continuous operator \(\Gamma \) called the total trace, from \(H^2(\Omega )\) to \(L^2(\partial \Omega )\times L^2(\partial \Omega )\) defined by \(\Gamma (u)=\left( \gamma _0(u),\gamma _1(u)\right) \), where \(\gamma _0(u)\) is the trace of u on \(\partial \Omega \) and \(\gamma _1(u)\) is the normal derivative of u. In particular, for \(u\in C^{2}(\overline{\Omega })\), \(\gamma _0(u)=u_{|_{\partial \Omega }}\) and \(\gamma _1(u)=\frac{\partial u}{\partial \nu }=\nabla u_{|_{\partial \Omega }}\cdot \nu \), where \(\nu \) denotes the outer unit normal to \(\partial \Omega \).

A relevant problem in the theory of Sobolev Spaces consists in describing the trace spaces \(\gamma _0(H^2(\Omega ))\), \(\gamma _1(H^2(\Omega ))\), and the total trace space \(\Gamma (H^2(\Omega ))\). This problem has important implications in the study of solutions to fourth order elliptic partial differential equations.

From a historical point of view, this issue finds its origins in [23] where J. Hadamard proposed his famous counterexample pointing out the importance to understand which conditions on the datum g guarantee that the solution v to the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{ll}\Delta v=0,&{} \mathrm{in}\ \Omega ,\\ v=g,&{} \mathrm{on }\ \partial \Omega , \end{array} \right. \end{aligned}$$

has square summable gradient. In modern terms, this problem can be reformulated as the problem of finding necessary and sufficient conditions on g such that \(g=\gamma _0(u)\) for some \(u\in H^1(\Omega )\).

If the domain \(\Omega \) is of class \(C^{2,1}\), then it is known that \(\gamma _0(H^2(\Omega ))=H^{\frac{3}{2}}(\partial \Omega )\), \(\gamma _1(H^2(\Omega ))=H^{\frac{1}{2}}(\partial \Omega )\), and \(\Gamma (H^2(\Omega ))=H^{\frac{3}{2}}(\partial \Omega )\times H^{\frac{1}{2}}(\partial \Omega )\), where \(H^{\frac{3}{2}}(\partial \Omega )\) and \(H^{\frac{1}{2}}(\partial \Omega )\) are the classical Sobolev spaces of fractional order (see e.g., [22, 31] for their definitions). However, if \(\Omega \) is an arbitrary bounded domain of class \(C^{0,1}\) there is no such a simple description and not many results are available in the literature.

We note that a complete description of the traces of all derivatives up to the order \(m-1\) of a function \(u\in H^m(\Omega )\) is due to O. Besov who provided an explicit but quite technical representation theorem, see [6, 7], see also [8]. Simpler descriptions are not available with the exception of a few special cases. For example, when \(\Omega \) is a polygon in \(\mathbb R^2\) the trace spaces are described by using the classical trace theorem applied to each side of the polygon, complemented with suitable compatibility conditions at the vertexes, see [22] also for higher dimensional polyhedra. For more general planar domains another simple description is given in [20].

Our list of references cannot be exhaustive and we refer to the recent monograph [30] which treats the trace problem in presence of corner or conical singularities in \(\mathbb R^3\), as well as further results on N-dimensional polyhedra. We also quote the fundamental paper  [24] by V. Kondrat’ev for a pioneering work in this type of problems. Interested readers can find more information in our recent survey paper [26].

Thus, the definition of the space \(H^{\frac{3}{2}}(\partial \Omega )\) turns out to be problematic and for this reason sometimes the space \(H^{\frac{3}{2}}(\partial \Omega )\) is simply defined by setting \(H^{\frac{3}{2}}(\partial \Omega ):=\gamma _0(H^2(\Omega ))\) without providing an explicit representation. Note that standard definitions of \(H^s(\partial \Omega )\) when \(s\in (1,2]\) require that \(\Omega \) is of class at least \(C^{2}\).

In the present paper we provide decompositions of the space \(H^2(\Omega )\) of the form \(H^2(\Omega )=H^2_{\mu ,D}(\Omega )+\mathcal {H}^2_{0,N}(\Omega )\) and \(H^2(\Omega )=H^2_{\lambda ,N}(\Omega )+\mathcal {H}^2_{0,D}(\Omega )\). The spaces \(\mathcal {H}^2_{0,N}(\Omega )\) and \(\mathcal {H}^2_{0,D}(\Omega )\) are the subspaces of \(H^2(\Omega )\) of those functions u such that \(\gamma _1(u)=0\) and \(\gamma _0(u)=0\), respectively. The spaces \(H^2_{\mu ,D}(\Omega )\) and \(H^2_{\lambda ,N}(\Omega )\) are associated with suitable Steklov problems of biharmonic type (namely, problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) described here below), depending on real parameters \(\mu ,\lambda \), and admit Fourier bases of Steklov eigenfunctions, see (3.5) and (3.15). Under the sole assumptions that \(\Omega \) is of class \(C^{0,1}\), we use those bases to define in a natural way two spaces at the boundary which we denote by \(\mathcal S^{\frac{3}{2}}(\partial \Omega )\) and \(\mathcal S^{\frac{1}{2}}(\partial \Omega )\) and we prove that

$$\begin{aligned} \gamma _0(H^2(\Omega ))=\gamma _0(H^2_{\lambda ,N}(\Omega ))=\mathcal S^{\frac{3}{2}}(\partial \Omega ) \end{aligned}$$

and

$$\begin{aligned} \gamma _1(H^2(\Omega ))=\gamma _1(H^2_{\mu ,D}(\Omega ))=\mathcal S^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

see Theorem 4.1. Thus, if one would like to define the space \(H^\frac{3}{2}(\partial \Omega )\) as \(\gamma _0(H^2(\Omega ))\), our result gives an explicit description of \(H^\frac{3}{2}(\partial \Omega )\).

It turns out that the analysis of problems (\(\mathrm{BS}_{\mu }\))-(\(\mathrm{BS}_{\lambda }\)) provides further information on the total trace \(\Gamma (H^2(\Omega ))\). In particular, we prove the inclusion \(\Gamma (H^2(\Omega ))\subseteq \mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega )\) and show that in general this inclusion is strict if \(\Omega \) is assumed to be only of class \(C^{0,1}\). Moreover, we show that any couple \((f,g)\in \mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega )\) belongs to \(\Gamma (H^2(\Omega ))\) if and only if it satisfies a certain compatibility condition, see Theorem 4.4.

If \(\Omega \) is of class \(C^{2,1}\), we recover the classical result, namely \(\Gamma (H^2(\Omega ))=\mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega )\), which implies that \(\mathcal S^{\frac{3}{2}}(\partial \Omega )=H^{\frac{3}{2}}(\partial \Omega )\) and \(\mathcal S^{\frac{1}{2}}(\partial \Omega )=H^{\frac{1}{2}}(\partial \Omega )\).

The two families of problems which we are going to introduce depend on a parameter \(\sigma \in \big (-\frac{1}{N-1},1\big )\), which in applications to linear elasticity represents the Poisson coefficient of the elastic material of the underlying system for \(N=2\).

The first family of \(\mathrm{BS}_{\mu }\) - ‘Biharmonic Steklov \(\mu \)’ problems is defined as follows:

figurea

in the unknowns \(v,\lambda (\mu )\), where \(\mu \in \mathbb R\) is fixed. Here \(D^2 u\) denotes the Hessian matrix of u, \(\mathrm{div}_{\partial \Omega }F:=\mathrm{div}F-(\nabla F\cdot \nu )\nu \) denotes the tangential divergence of a vector field F and \(F_{\partial \Omega }:=F-(F\cdot \nu )\nu \) denotes the tangential component of F.

The second family of \(\mathrm{BS}_{\lambda }\) - ‘Biharmonic Steklov \(\lambda \)’ problems is defined as follows:

figureb

in the unknowns \(u,\mu (\lambda )\), where \(\lambda \in \mathbb R\) is fixed.

Note that since \(\Omega \) is assumed to be of class \(C^{0,1}\), problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) have to be considered in the weak sense, see (3.4) and (3.14) for the appropriate formulations.

Up to our knowledge, the Steklov problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) are new in the literature. Other Steklov-type problems for the biharmonic operator have been discussed in the literature. We mention the DBS - ‘Dirichlet Biharmonic Steklov’ problem

figurec

in the unknowns \(v,\eta \), and the NBS - ‘Neumann Biharmonic Steklov’ problem

figured

in the unknowns \(u,\xi \). Problem (\(\mathrm{DBS}\)) for \(\sigma =1\) has been studied by many authors (see e.g., [3, 9, 16,17,18, 25, 29]); for the case \(\sigma \ne 1\) we refer to [10], see also [4, 17] for \(\sigma =0\). Problem (\(\mathrm{NBS}\)) has been discussed in [25, 28, 29] for \(\sigma =1\). We point out that problem (\(\mathrm{BS}_{\lambda }\)) with \(\sigma =\lambda =0\) has been introduced in [12] as the natural fourth order generalization of the classical Steklov problem for the Laplacian (see also [11]). As we shall see, problem (\(\mathrm{BS}_{\lambda }\)) shares much more analogies with the classical Steklov problem than those already presented in [12], in particular it plays a role in describing the space \(\gamma _0(H^2(\Omega ))\) similar to that played by the Steklov problem for the Laplacian in describing \(\gamma _0(H^1(\Omega ))\) (cf. [2]).

If \(\mu <0\), problem (\(\mathrm{BS}_{\mu }\)) has a discrete spectrum which consists of a divergent sequence \(\left\{ \lambda _j(\mu )\right\} _{j=1}^{\infty }\) of non-negative eigenvalues of finite multiplicity. Similarly, if \(\lambda <\eta _1\), where \(\eta _1>0\) is the first eigenvalue of (\(\mathrm{DBS}\)), problem (\(\mathrm{BS}_{\lambda }\)) has a discrete spectrum which consists of a divergent sequence \(\left\{ \mu _j(\lambda )\right\} _{j=1}^{\infty }\) of eigenvalues of finite multiplicity and bounded from below. (For other values of \(\mu \) and \(\lambda \) the description of the spectra of (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) is more involved, see Appendix C.)

The eigenfunctions associated with the eigenvalues \(\lambda _j(\mu )\) define a Hilbert basis of the above mentioned space \(H^2_{\mu ,D}(\Omega )\) which is the orthogonal complement in \(H^2(\Omega )\) of \(\mathcal H^2_{0,N}(\Omega )\) with respect to a suitable scalar product. Moreover, the normal derivatives of those eigenfunctions allow to define the above mentioned space \(\mathcal S^{\frac{1}{2}}(\partial \Omega )\), see (4.2). Similarly, the eigenfunctions associated with the eigenvalues \(\mu _j(\lambda )\) define a Hilbert basis of the space \(H^2_{\lambda ,N}(\Omega )\) which is the orthogonal complement in \(H^2(\Omega )\) of \(\mathcal H^2_{0,D}(\Omega )\) with respect to a suitable scalar product. Moreover, the traces of those eigenfunctions allow to define the space \(\mathcal S^{\frac{3}{2}}(\partial \Omega )\), see (4.1).

The definitions in (4.1) and (4.2) are given by means of Fourier series and the coefficients in such expansions need to satisfy certain summability conditions, which are strictly related to the asymptotic behavior of the eigenvalues of (\(\mathrm{BS}_{\lambda }\)) and (\(\mathrm{BS}_{\mu }\)). Note that

$$\begin{aligned} \mu _j(\lambda )\sim C_N\left( \frac{j}{|\partial \Omega |}\right) ^\frac{3}{N-1}\mathrm{\ \ \ and\ \ \ }\lambda _j(\mu )\sim C_N'\left( \frac{j}{|\partial \Omega |}\right) ^\frac{1}{N-1}\,,\mathrm{\ as\ }j\rightarrow +\infty , \end{aligned}$$
(1.1)

where \(C_N,C_N'\) depend only on N, see Appendix B. In view of (1.1) and (4.1)–(4.2), we can identify the space \(\mathcal S^{\frac{3}{2}}(\partial \Omega )\) with the space of sequences

$$\begin{aligned} \left\{ (s_j)_{j=1}^{\infty }\in \mathbb R^{\infty }:(j^{\frac{3}{2(N-1)}}s_j)_{j=1}^{\infty }\in l^2\right\} \end{aligned}$$
(1.2)

and the space \(\mathcal S^{\frac{1}{2}}(\partial \Omega )\) with the space

$$\begin{aligned} \left\{ (s_j)_{j=1}^{\infty }\in \mathbb R^{\infty }:(j^{\frac{1}{2(N-1)}}s_j)_{j=1}^{\infty }\in l^2\right\} . \end{aligned}$$
(1.3)

Observe the natural appearance of the exponents \(\frac{3}{2}\) and \(\frac{1}{2}\) in (1.2) and (1.3). It is remarkable that, in essence, a summability condition analogous to that in (1.3) is already present in [23, Formula (3)] for the case of the unit disk D of the plane and the space \(H^{\frac{1}{2}}(\partial D)=\gamma _0(H^1(D))\).

Using the representations (4.1) and (4.2) we are able to provide necessary and sufficient conditions for the solvability in \(H^2(\Omega )\) of the Dirichlet problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2 u=0\,, &{} \mathrm{in\ }\Omega ,\\ u=f\,, &{} \mathrm{on\ }\partial \Omega ,\\ \frac{\partial u}{\partial \nu }=g\,, &{} \mathrm{on\ }\partial \Omega , \end{array}\right. } \end{aligned}$$
(1.4)

under the sole assumption that \(\Omega \) is of class \(C^{0,1}\), and to represent in Fourier series the solutions. We note that different necessary and sufficient conditions for the solvability of problem (1.4) in the larger space \(H(\Delta ,\Omega )=\left\{ u\in H^1(\Omega ):\Delta u\in L^2(\Omega )\right\} \) have been found in [4] by using the (\(\mathrm{DBS}\)) problem with \(\sigma =1\) and the classical Dirichlet-to-Neumann map. We also refer to [5, 15, 34, 35] for a different approach to the solvability of higher order problems on Lipschitz domains. Note that in [5, 15, 34, 35] the authors consider notions of weak solutions which differ substantially from the standard variational one used in this paper, and the solutions to problem (1.4) are allowed to have infinite Dirichlet energy. For instance, in [15, Thm. 3.1] the boundary data fg belong to \(L^2_1(\partial \Omega ), L^2(\partial \Omega ) \) respectively, and are assumed by the solution u as non-tangential limits; accordingly, u is not expected to belong to \(H^{2}(\Omega )\) but just to \(H^{3/2}(\Omega )\), see [15, p. 110].

Since we have not been able to find problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) in the literature, we believe that it is worth including in the present paper also some information on their spectral behavior, which may have a certain interest on its own. In particular, we prove Lipschitz continuity results for the functions \(\mu \mapsto \lambda _j(\mu )\) and \(\lambda \mapsto \mu _j(\lambda )\) and we show that problems (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)) can be seen as limiting problems for (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) as \(\mu \rightarrow -\infty \) and \(\lambda \rightarrow -\infty \), respectively. We also perform a complete study of the eigenvalues in the unit ball in \(\mathbb R^N\) for \(\sigma =0\), and we discuss the asymptotic behavior of \(\lambda _j(\mu )\) and \(\mu _j(\lambda )\) on smooth domains when \(j\rightarrow +\infty \). Finally, we briefly discuss problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) also when \(\mu > 0\) and \(\lambda > \eta _1\).

Our approach is similar to that developed by G. Auchmuty in [2] for the trace space of \(H^1(\Omega )\), based on the classical second order Steklov problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta u=0\,, &{} \mathrm{in\ }\Omega ,\\ \frac{\partial u}{\partial \nu }=\lambda u\,, &{} \mathrm{on\ }\partial \Omega . \end{array}\right. } \end{aligned}$$

We also refer to [32] for related results.

This paper is organized as follows. In Section 2 we introduce some notation and discuss a few preliminary results. In Section 3 we discuss problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) when \(\mu <0\) and \(\lambda <\eta _1\). In Section 4 we define the spaces \(\mathcal S^{\frac{3}{2}}(\partial \Omega )\) and \(\mathcal S^{\frac{1}{2}}(\partial \Omega )\) and the representation theorems for the trace spaces of \(H^2(\Omega )\). In Subsect. 4.1 we prove a representation result for the solutions of the biharmonic Dirichlet problem. In Appendix A we provide a complete description of problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) on the unit ball for \(\sigma =0\). In Appendix B we briefly discuss asymptotic laws for the eigenvalues. In Appendix C we discuss problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) when \(\mu > 0\) and \(\lambda >\eta _1\).

Preliminaries and notation

For a bounded domain (i.e., a bounded open connected set) \(\Omega \) in \(\mathbb R^N\), we denote by \(H^1(\Omega )\) the standard Sobolev space of functions in \(L^2(\Omega )\) with all weak derivatives of the first order in \(L^2(\Omega )\) endowed with its standard norm \(\Vert u\Vert _{H^1(\Omega )}:=\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}+\Vert u\Vert ^2_{L^2(\Omega )}\right) ^{\frac{1}{2}}\) for all \(u\in H^1(\Omega )\). Note that in this paper we consider \(L^2(\Omega )\) as a space of real-valued functions and we always assume \(N\ge 2\).

By \(H^2(\Omega )\) we denote the standard Sobolev space of functions in \(L^2(\Omega )\) with all weak derivatives of the first and second order in \(L^2(\Omega )\) endowed with the norm \(\Vert u\Vert _{H^2(\Omega )}:=\left( \Vert D^2u\Vert ^2_{L^2(\Omega )}+\Vert u\Vert ^2_{L^2(\Omega )}\right) ^{\frac{1}{2}}\) for all \(u\in H^2(\Omega )\). We denote by \(H_0^1(\Omega )\) the closure of \(C^{\infty }_c(\Omega )\) in \(H^1(\Omega )\) and by \(H^2_0(\Omega )\) the closure of \(C^{\infty }_c(\Omega )\) in \(H^2(\Omega )\). The space \(C^{\infty }_c(\Omega )\) is the space of all functions in \(C^{\infty }(\Omega )\) with compact support in \(\Omega \). If the boundary is sufficiently regular (e.g., if \(\Omega \) is of class \(C^{0,1}\)), the norm defined by \(\sum _{|\alpha |\le 2}\Vert D^{\alpha }u\Vert _{L^2(\Omega )}\) is a norm on \(H^2(\Omega )\) equivalent to the standard one.

By definition, a domain of class \(C^{0,1}\) is such that locally around each point of its boundary it can be described as the sub-graph of a Lipschitz continuous function. Also, we shall say that \(\Omega \) is of class \(C^{2,1}\) if locally around each point of its boundary the domain can be described as the sub-graph of a function of class \(C^{2,1}\).

By \((\cdot ,\cdot )_{\partial \Omega }\) we denote the standard scalar product of \(L^2(\partial \Omega )\), namely

$$\begin{aligned} (u,v)_{\partial \Omega }:=\int _{\partial \Omega }uv d\sigma \,,\ \ \ \forall u,v\in L^2(\partial \Omega ). \end{aligned}$$

We denote by \(\gamma _0(u)\in L^2(\partial \Omega )\) the trace of u and by \(\gamma _1(u)\in L^2(\partial \Omega )\) the normal derivative of u, that is, \(\gamma _1(u)=\nabla u\cdot \nu \). By \(\Gamma \) we denote the total trace operator from \(H^2(\Omega )\) to \(L^2(\partial \Omega )\times L^2(\partial \Omega )\) defined by

$$\begin{aligned} \Gamma _2(u)=(\gamma _0(u),\gamma _1(u))\,, \end{aligned}$$

for all \(u\in H^2(\Omega )\). The operator \(\Gamma \) is compact. If \(\Omega \) is of class \(C^{2,1}\) then \(\Gamma \) is a linear and continuous operator from \(H^2(\Omega )\) onto \(H^{\frac{3}{2}}(\partial \Omega )\times H^{\frac{1}{2}}(\partial \Omega )\) admitting a right continuous inverse. We refer to e.g., [31] for more details.

Here \(H^{\frac{3}{2}}(\partial \Omega ), H^{\frac{1}{2}}(\partial \Omega )\) denote the standard Sobolev spaces of fractional order (see e.g., [22, 31] for more details). For any \(\sigma \in \big (-\frac{1}{N-1},1\big )\), \(\mu ,\lambda \in \mathbb R\) and \(u,v\in H^2(\Omega )\) we set

$$\begin{aligned} \mathcal Q_{\sigma }(u,v)= & {} (1-\sigma )\int _{\Omega }D^2u:D^2v dx+\sigma \int _{\Omega }\Delta u\Delta v dx,\\ \mathcal {Q}_{\mu ,D}(u,v)= & {} \mathcal Q_{\sigma }(u,v)-\mu (\gamma _0(u),\gamma _0(v))_{\partial \Omega }, \end{aligned}$$

and

$$\begin{aligned} \mathcal {Q}_{\lambda ,N}(u,v)=\mathcal Q_{\sigma }(u,v)-\lambda (\gamma _1(u),\gamma _1(v))_{\partial \Omega }, \end{aligned}$$

where \(D^2u:D^2 v=\sum _{i,j=1}^N\frac{\partial ^2u}{\partial x_i\partial x_j}\frac{\partial ^2 v}{\partial x_i\partial x_j}\) denotes the Frobenius product of the Hessians matrices. Note that if \(\sigma \in \big (-\frac{1}{N-1},1\big )\), then the quadratic form \(\mathcal Q_{\sigma }\) is coercive in \(H^2(\Omega )\) and the norm \(\left( \mathcal Q_{\sigma }(u,u)+\Vert u\Vert _{L^2(\Omega )}^2\right) ^{\frac{1}{2}}\) is equivalent to the standard norm of \(H^2(\Omega )\), see e.g., [14].

It is easy to see that if \(\Omega \) is a bounded domain of class \(C^{0,1}\) then the space \(H^2(\Omega )\) can be endowed with the equivalent norm

$$\begin{aligned} \left( \Vert D^2u\Vert ^2_{L^2(\Omega )}+\Vert \gamma _0(u)\Vert ^2_{L^2(\partial \Omega )}\right) ^{\frac{1}{2}}. \end{aligned}$$

We set

$$\begin{aligned} \mathcal {H}^2_{0,D}(\Omega )=\left\{ u\in H^2(\Omega ):\gamma _0(u)=0\right\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {H}^2_{0,N}(\Omega )=\left\{ u\in H^2(\Omega ):\gamma _1(u)=0\right\} . \end{aligned}$$

The spaces \(\mathcal {H}_{0,D}^2(\Omega )\) and \(\mathcal {H}_{0,N}^2(\Omega )\) are closed subspaces of \(H^2(\Omega )\) and \(\mathcal {H}^2_{0,N}(\Omega )\cap \mathcal {H}^2_{0,D}(\Omega )=H^2_0(\Omega )\). We also note that \(\mathcal {H}^2_{0,D}=H^2(\Omega )\cap H^1_0(\Omega )\).

It is useful to recall the so-called biharmonic Green formula

$$\begin{aligned}&\int _{\Omega }D^2\psi :D^2\varphi dx=\int _{\Omega }(\Delta ^2\psi )\varphi dx+\int _{\partial \Omega }\frac{\partial ^2\psi }{\partial \nu ^2}\frac{\partial \varphi }{\partial \nu }d\sigma \nonumber \\&\quad -\int _{\partial \Omega }\left( \mathrm{div}_{\partial \Omega }(D^2\psi \cdot \nu )_{\partial \Omega }+\frac{\partial \Delta \psi }{\partial \nu }\right) \varphi d\sigma , \end{aligned}$$
(2.1)

valid for all sufficiently smooth \(\psi ,\varphi \), see [1].

The biharmonic functions in \(H^2(\Omega )\) are defined as those functions \(u\in H^2(\Omega )\) such that \(\int _{\Omega }D^2u:D^2\varphi dx=0\) for all \(\varphi \in H^2_0(\Omega )\), or equivalently, thanks to (2.1), those functions \(u\in H^2(\Omega )\) such that \(\int _{\Omega }\Delta u\Delta \varphi dx=0\) for all \(\varphi \in H^2_0(\Omega )\) . We denote by \(\mathcal B_N(\Omega )\) the space of biharmonic functions with zero normal derivative, that is the orthogonal complement of \(H^2_0(\Omega )\) in \(\mathcal H^2_{0,N}(\Omega )\) with respect to \(\mathcal Q_{\sigma }\):

$$\begin{aligned} \mathcal B_N(\Omega ):=\left\{ u\in \mathcal H^2_{0,N}(\Omega ):\mathcal Q_{\sigma }(u,\varphi )=0\,,\forall \varphi \in H^2_0(\Omega )\right\} . \end{aligned}$$
(2.2)

By formula (2.1) and a standard approximation we deduce that

$$\begin{aligned} \mathcal B_N(\Omega ):=\left\{ u\in \mathcal H^2_{0,N}(\Omega ):\int _{\Omega }\Delta u\Delta \varphi dx=0\,,\forall \varphi \in H^2_0(\Omega )\right\} . \end{aligned}$$
(2.3)

We note that \(\mathcal B_N(\Omega )\) is the space of the biharmonic functions in \(\mathcal H^2_{0,N}(\Omega )\). Thus

$$\begin{aligned} \mathcal H^2_{0,N}(\Omega )= H^2_0(\Omega )\oplus \mathcal B_N(\Omega ). \end{aligned}$$

Analogously, we denote by \(\mathcal B_D\) the space of biharmonic functions with zero boundary trace, that is the orthogonal complement of \(H^2_0(\Omega )\) in \(\mathcal H^2_{0,D}(\Omega )\) with respect to \(\mathcal Q_{\sigma }\):

$$\begin{aligned} \mathcal B_D(\Omega ):=\left\{ u\in \mathcal H^2_{0,D}(\Omega ):\mathcal Q_{\sigma }(u,\varphi )=0\,,\forall \varphi \in H^2_0(\Omega )\right\} . \end{aligned}$$
(2.4)

By formula (2.1) and standard approximation we deduce that

$$\begin{aligned} \mathcal B_D(\Omega ):=\left\{ u\in \mathcal H^2_{0,D}(\Omega ):\int _{\Omega }\Delta u\Delta \varphi dx=0\,,\forall \varphi \in H^2_0(\Omega )\right\} . \end{aligned}$$
(2.5)

We note that \(\mathcal B_D(\Omega )\) is the space of biharmonic functions in \(\mathcal H^2_{0,D}(\Omega )\). Thus

$$\begin{aligned} \mathcal H^2_{0,D}(\Omega )= H^2_0(\Omega )\oplus \mathcal B_D(\Omega ). \end{aligned}$$

Finally, by \(\mathbb N\) we denote the set of positive natural numbers and by \(\mathbb N_0\) the set \(\mathbb N\,\cup \,\left\{ 0\right\} \).

Multi-parameter Steklov problems

In this section we provide the appropriate weak formulations of problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)). In particular we prove that both problems have discrete spectrum provided \(\mu <0\) and \(\lambda <\eta _1\), respectively. Here \(\eta _1\) is the first eigenvalue of problem (3.1) below, which is the weak formulation of (\(\mathrm{DBS}\)). We remark that \(\eta _1>0\) and that \(\xi _1=0\) is the first eigenvalue of problem (\(\mathrm{NBS}\)), hence the condition \(\mu <0\) reads \(\mu <\xi _1\). We also provide a variational characterization of the eigenvalues.

Through all this section \(\Omega \) will be a bounded domain of class \(C^{0,1}\) and \(\sigma \in \big (-\frac{1}{N-1},1\big )\) will be fixed.

The (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)) problems

Before analyzing problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) we need to recall a few facts about problems (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)).

Problem (\(\mathrm{DBS}\)) is understood in the weak sense as follows:

$$\begin{aligned} \int _{\Omega }(1-\sigma )D^2 v:D^2\varphi +\sigma \Delta v\Delta \varphi dx=\eta \int _{\partial \Omega }\frac{\partial v}{\partial \nu }\frac{\partial \varphi }{\partial \nu } d\sigma \,,\ \ \ \forall \varphi \in \mathcal {H}^2_{0,D}(\Omega ), \end{aligned}$$
(3.1)

in the unknowns \(v\in \mathcal {H}^2_{0,D}(\Omega )\), \(\eta \in \mathbb R\). Note that formulation (3.1) is justified by formula (2.1). Indeed, by applying formula (2.1), one can easily see that if v is a smooth solution to problem (3.1), then v is a solution to the classical problem (\(\mathrm{DBS}\)) (the same considerations can be done for all other problems discussed in this paper).

We have the following theorem.

Theorem 3.1

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\) and let \(\sigma \in \big (-\frac{1}{N-1},1\big )\). The eigenvalues of problem (3.1) have finite multiplicity and are given by a non-decreasing sequence of positive real numbers \(\eta _j\) defined by

$$\begin{aligned} \eta _j=\min _{\begin{array}{c} V\subset \mathcal {H}^2_{0,D}(\Omega )\\ \mathrm{dim}V=j \end{array}}\max _{\begin{array}{c} v\in V\\ u\ne 0 \end{array}}\frac{\mathcal Q_{\sigma }(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }, \end{aligned}$$
(3.2)

where each eigenvalue is repeated according to its multiplicity. Moreover, there exists a Hilbert basis \(\left\{ v_j\right\} _{j=1}^{\infty }\) of \(\mathcal B_D(\Omega )\) of eigenfunctions \(v_j\) associated with the eigenvalues \(\eta _j\). Finally, by normalizing the eigenfunctions \(v_j\) with respect to \(\mathcal Q_{\sigma }\) for all \(j\ge 1\), the functions \(\hat{v}_j:=\sqrt{\eta _j}\gamma _1(v_j)\) define a Hilbert basis of \(L^2(\partial \Omega )\) with respect to its standard scalar product.

Problem (\(\mathrm{NBS}\)) is understood in the weak sense as follows:

$$\begin{aligned} \int _{\Omega }(1-\sigma )D^2 u:D^2\varphi +\sigma \Delta u\Delta \varphi dx=\xi \int _{\partial \Omega }u\varphi d\sigma \,,\ \ \ \forall \varphi \in \mathcal {H}^2_{0,N}(\Omega ), \end{aligned}$$
(3.3)

in the unknowns \(u\in \mathcal {H}^2_{0,N}(\Omega )\), \(\xi \in \mathbb R\). We have the following theorem.

Theorem 3.2

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\) and let \(\sigma \in \big (-\frac{1}{N-1},1\big )\). The eigenvalues of problem (3.3) have finite multiplicity and are given by a non-decreasing sequence of non-negative real numbers \(\xi _j\) defined by

$$\begin{aligned} \xi _j=\min _{\begin{array}{c} U\subset \mathcal {H}^2_{0,N}(\Omega )\\ \mathrm{dim}U=j \end{array}}\max _{\begin{array}{c} u\in U\\ u\ne 0 \end{array}}\frac{\mathcal Q_{\sigma }(u,u)}{\int _{\partial \Omega }u^2d\sigma }, \end{aligned}$$

where each eigenvalue is repeated according to its multiplicity. The first eigenvalue \(\xi _1=0\) has multiplicity one and the corresponding eigenfunctions are the constant functions on \(\Omega \). Moreover, there exists a Hilbert basis \(\left\{ u_j\right\} _{j=1}^{\infty }\) of \(\mathcal B_N(\Omega )\) of eigenfunctions \(u_j\) associated with the eigenvalues \(\xi _j\). Finally, by normalizing the eigenfunctions \(u_j\) with respect to \(\mathcal Q_{\sigma }\) for all \(j\ge 2\), the functions \(\hat{u}_j:=\sqrt{\xi _j}\gamma _0(u_j)\), \(j\ge 2\), and \(\hat{u}_1=|\partial \Omega |^{-1/2}\) define a Hilbert basis of \(L^2(\partial \Omega )\) with respect to its standard scalar product.

The proofs of Theorems 3.1 and 3.2 can be carried out exactly as those of Theorems 3.3 and 3.10 presented in Subsects. 3.2 and 3.3. Note that the condition \(\sigma \in \big (-\frac{1}{N-1},1\big )\) is used to guarantee the coercivity of the form \(\mathcal Q_{\sigma }\) discussed in the previous section.

The \(BS_{\mu }\) eigenvalue problem

For any \(\mu \in \mathbb R\), the weak formulation of problem (\(\mathrm{BS}_{\mu }\)) reads

$$\begin{aligned}&\int _{\Omega }(1-\sigma )D^2 v:D^2\varphi +\sigma \Delta v\Delta \varphi dx-\mu \int _{\partial \Omega }v\varphi d\sigma \nonumber \\&=\lambda (\mu )\int _{\partial \Omega }\frac{\partial v}{\partial \nu }\frac{\partial \varphi }{\partial \nu } d\sigma \,,\ \ \ \forall \varphi \in H^2(\Omega ), \end{aligned}$$
(3.4)

in the unknowns \(v\in H^2(\Omega )\), \(\lambda (\mu )\in \mathbb R\), and can be re-written as

$$\begin{aligned} \mathcal {Q}_{\mu ,D}(v,\varphi )=\lambda (\mu )\left( \gamma _1(v),\gamma _1(\varphi )\right) _{\partial \Omega }\,,\ \ \ \forall \varphi \in H^2(\Omega ). \end{aligned}$$

We prove that for all \(\mu <0\), problem (\(\mathrm{BS}_{\mu }\)) admits an increasing sequence of eigenvalues of finite multiplicity diverging to \(+\infty \) and the corresponding eigenfunctions form a basis of \(H^2_{\mu ,D}(\Omega )\), where \(H^2_{\mu ,D}(\Omega )\) denotes the orthogonal complement of \(\mathcal {H}^2_{0,N}(\Omega )\) in \(H^2(\Omega )\) with respect to the scalar product \(\mathcal {Q}_{\mu ,D}\), namely

$$\begin{aligned} H^2_{\mu ,D}(\Omega )=\left\{ v\in H^2(\Omega ):\mathcal {Q}_{\mu ,D}(v,\varphi )=0\,,\ \forall \varphi \in \mathcal {H}^2_{0,N}(\Omega )\right\} . \end{aligned}$$
(3.5)

To do so, we recast problem (3.4) in the form of an eigenvalue problem for a compact self-adjoint operator acting on a Hilbert space. We consider on \(H^2(\Omega )\) the equivalent norm

$$\begin{aligned} \Vert v\Vert _{\mu ,D}^2=\mathcal {Q}_{\mu ,D}(v,v) \end{aligned}$$

which is associated with the scalar product defined by

$$\begin{aligned} \langle v,\varphi \rangle _{\mu ,D}=\mathcal {Q}_{\mu ,D}(v,\varphi ), \end{aligned}$$

for all \(v,\varphi \in H^2(\Omega )\). Then we define the operator \(B_{\mu ,D}\) from \(H^2(\Omega )\) to its dual \((H^2(\Omega ))'\) by setting

$$\begin{aligned} B_{\mu ,D}(v)[\varphi ]=\langle v,\varphi \rangle _{\mu ,D}\,,\ \ \ \forall v,\varphi \in H^2(\Omega ). \end{aligned}$$

By the Riesz Theorem it follows that \(B_{\mu ,D}\) is a surjective isometry. Then we consider the operator \(J_1\) from \(H^2(\Omega )\) to \((H^2(\Omega ))'\) defined by

$$\begin{aligned} J_1(v)[\varphi ]=(\gamma _1(v),\gamma _1(\varphi ))_{\partial \Omega }\,,\ \ \ \forall v,\varphi \in H^2(\Omega ). \end{aligned}$$
(3.6)

The operator \(J_1\) is compact since \(\gamma _1\) is a compact operator from \(H^2(\Omega )\) to \(L^2(\partial \Omega )\). Finally, we set

$$\begin{aligned} T_{\mu ,D}=B_{\mu ,D}^{(-1)}\circ J_1. \end{aligned}$$
(3.7)

From the compactness of \(J_1\) and the boundedness of \(B_{\mu ,D}\) it follows that \(T_{\mu ,D}\) is a compact operator from \(H^2(\Omega )\) to itself. Moreover, \(\langle T_{\mu ,D}(v),\varphi \rangle _{\mu ,D}=(\gamma _1(v),\gamma _1(\varphi ))_{\partial \Omega }\), for all \(u,\varphi \in H^2(\Omega )\), hence \(T_{\mu ,D}\) is self-adjoint. Note that \(\mathrm{Ker}\,T_{\mu ,D}=\mathrm{Ker}\,J_1=\mathcal {H}^2_{0,N}(\Omega )\) and the non-zero eigenvalues of \(T_{\mu ,D}\) coincide with the reciprocals of the eigenvalues of (3.4), the eigenfunctions being the same.

We are now ready to prove the following theorem.

Theorem 3.3

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\) and let \(\sigma \in \big (-\frac{1}{N-1},1\big )\). Let \(\mu <0\). Then the eigenvalues of (3.4) have finite multiplicity and are given by a non-decreasing sequence of positive real numbers \(\left\{ \lambda _j(\mu )\right\} _{j=1}^{\infty }\) defined by

$$\begin{aligned} \lambda _j(\mu )=\min _{\begin{array}{c} V\subset H^2(\Omega )\\ \mathrm{dim}V=j \end{array}}\max _{\begin{array}{c} v\in V\\ \frac{\partial v}{\partial \nu }\ne 0 \end{array}}\frac{\mathcal {Q}_{\mu ,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }, \end{aligned}$$
(3.8)

where each eigenvalue is repeated according to its multiplicity.

Moreover there exists a basis \(\left\{ v_{j,\mu }\right\} _{j=1}^{\infty }\) of \(H^2_{\mu ,D}(\Omega )\) of eigenfunctions \(v_{j,\mu }\) associated with the eigenvalues \(\lambda _j(\mu )\).

By normalizing the eigenfunctions \(v_{j,\mu }\) with respect to \(\mathcal {Q}_{\mu ,D}\), the functions defined by \(\left\{ \hat{v}_{j,\mu }\right\} _{j=1}^{\infty }:=\left\{ \sqrt{\lambda _{j}(\mu )}\gamma _1(v_{j,\mu })\right\} _{j=1}^{\infty }\) form a Hilbert basis of \(L^2(\partial \Omega )\) with respect to its standard scalar product.

Proof

Since \(\mathrm{Ker}\, T_{\mu ,D}=\mathcal H^2_{0,N}\), by the Hilbert-Schmidt Theorem applied to the compact self-adjoint operator \(T_{\mu ,D}\) it follows that \(T_{\mu ,D}\) admits an increasing sequence of positive eigenvalues \(\left\{ q_j\right\} _{j=1}^{\infty }\) bounded from above, converging to zero and a corresponding Hilbert basis \(\left\{ v_{j,\mu }\right\} _{j=1}^{\infty }\) of eigenfunctions of \(H^2_{\mu ,D}(\Omega )\). Since \(q\ne 0\) is an eigenvalue of \(T_{\mu ,D}\) if and only if \(\lambda =\frac{1}{q}\) is an eigenvalue of (3.4) with the same eigenfunctions, we deduce the validity of the first part of the statement. In particular, formula (3.8) follows from the standard min-max formula for the eigenvalues of compact self-adjoint operators. Note that \(\lambda _1(\mu )>0\), since \(Q_{\mu ,D}(v,v)=0\) if and only if \(v=0\).

To prove the final part of the theorem, we recast problem (3.4) into an eigenvalue problem for the compact self-adjoint operator \(T_{\mu ,D}'=\gamma _1\circ B_{\mu ,D}^{(-1)}\circ J_1'\), where \(J_1'\) denotes the map from \(L^2(\partial \Omega )\) to the dual of \(H^2(\Omega )\) defined by

$$\begin{aligned} J_1'(v)[\varphi ]=(v,\gamma _1(\varphi ))_{\partial \Omega }\,,\ \ \ \forall v\in L^2(\partial \Omega ),\varphi \in H^2(\Omega ). \end{aligned}$$

We apply again the Hilbert-Schmidt Theorem and observe that \(T_{\mu ,D}\) and \(T_{\mu ,D}'\) admit the same non-zero eigenvalues and that the eigenfunctions of \(T_{\mu ,D}'\) are exactly the normal derivatives of the eigenfunctions of \(T_{\mu ,D}\). From (3.4) we deduce that if the eigenfunctions \(v_{j,\mu }\) of \(T_{\mu ,D}\) are normalized by \(\mathcal {Q}_{\mu ,D}(v_{j,\mu },v_{k,\mu })=\delta _{jk}\), where \(\delta _{jk}\) is the Kronecker symbol, then the normalization of the traces of their normal derivatives in \(L^2(\partial \Omega )\) are obtained by multiplying \(\gamma _1(v_{j,\mu })\) by \(\sqrt{\lambda _j(\mu )}\). This concludes the proof.

\(\square \)

We present now a few results on the behavior of the eigenvalues of (3.4) for \(\mu \in (-\infty ,0)\), in particular we prove a Lipschitz continuity result for the eigenvalues \(\lambda _j(\mu )\) with respect to \(\mu \) and find their limits as \(\mu \rightarrow -\infty \).

Theorem 3.4

For any \(j\in \mathbb N\) and \(\delta >0\), the function \(\lambda _j:(-\infty ,-\delta ]\rightarrow (0,+\infty )\) which takes \(\mu \in (-\infty ,-\delta ]\) to \(\lambda _j(\mu )\in (0,+\infty )\) is Lipschitz continuous on \((-\infty ,-\delta ]\).

Proof

Without loss of generality we assume that \(\mu _1,\mu _2\in (-\infty ,-\delta ]\) and that \(\mu _1<\mu _2\). Let \(v\in H^2(\Omega )\). Then

$$\begin{aligned} 0\le & {} \frac{\mathcal Q_{\mu _1,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }-\frac{\mathcal Q_{\mu _2,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }=(\mu _2-\mu _1)\frac{\int _{\partial \Omega }v^2d\sigma }{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }\\\le & {} -\frac{(\mu _2-\mu _1)}{\mu _1}\frac{\mathcal Q_{\mu _1,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }. \end{aligned}$$

Hence

$$\begin{aligned} \frac{\mathcal Q_{\mu _1,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }\ge \frac{\mathcal Q_{\mu _2,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma } \end{aligned}$$
(3.9)

and

$$\begin{aligned} \frac{\mathcal Q_{\mu _2,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }\ge \frac{\mathcal Q_{\mu _1,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }\left( 1+\frac{(\mu _2-\mu _1)}{\mu _1}\right) \end{aligned}$$
(3.10)

By taking the infimum and the supremum over j dimensional subspaces of \(H^2(\Omega )\) into (3.9) and (3.10), and by (3.8), we get

$$\begin{aligned} |\lambda _j(\mu _1)-\lambda _j(\mu _2)|\le \frac{\lambda _j(\mu _1)}{|\mu _1|}|\mu _2-\mu _1|\le \lambda _j(\mu _1)\frac{|\mu _2-\mu _1|}{\delta }. \end{aligned}$$

This concludes the proof. \(\square \)

We now investigate the behavior of the eigenvalues \(\lambda _j(\mu )\) as \(\mu \rightarrow -\infty \). First, we need to recall a few facts about the convergence of operators defined on variable spaces. As customary, we consider families of spaces and operators depending on a small parameter \(\varepsilon \ge 0\) with \(\varepsilon \rightarrow 0\). This will be applied later with \(\varepsilon =-\frac{1}{\mu }\) and \(\mu \rightarrow -\infty \).

Let us denote by \(\mathcal H_{\varepsilon }\) a family of Hilbert spaces for all \(\varepsilon \in [0,\varepsilon _0)\) and assume that there exists a corresponding family of linear operators \(E_{\varepsilon }:\mathcal H_0\rightarrow \mathcal H_{\varepsilon }\) such that, for any \(u\in \mathcal H_0\)

$$\begin{aligned} \Vert E_{\varepsilon }(u)\Vert _{\mathcal H_{\varepsilon }}\rightarrow \Vert u\Vert _{\mathcal H_0}\,,\ \ \ \mathrm{as\ }\varepsilon \rightarrow 0. \end{aligned}$$

We recall the definition of compact convergence of operators in the sense of [33].

Definition 3.5

We say that a family \(\left\{ K_{\varepsilon }\right\} _{\varepsilon \in [0,\varepsilon _0)}\) of compact operators \(K_{\varepsilon }\in \mathcal L(\mathcal H_{\varepsilon })\) converges compactly to \(K_0\) if

  1. i)

    for any \(\left\{ u_{\varepsilon }\right\} _{\varepsilon \in (0,\varepsilon _0)}\) with \(\Vert u_{\varepsilon }-E_{\varepsilon }(u)\Vert _{\mathcal H_{\varepsilon }}\rightarrow 0\) as \(\varepsilon \rightarrow 0\), then \(\Vert K_{\varepsilon }(u_{\varepsilon })-E_{\varepsilon }(K_0(u))\Vert _{\mathcal H_{\varepsilon }}\rightarrow 0\) as \(\varepsilon \rightarrow 0\);

  2. ii)

    for any \(\left\{ u_{\varepsilon }\right\} _{\varepsilon \in (0,\varepsilon _0)}\) with \(u_{\varepsilon }\in \mathcal H_{\varepsilon }\), \(\Vert u_{\varepsilon }\Vert _{\mathcal H_{\varepsilon }}=1\), then \(\left\{ K_{\varepsilon }(u_{\varepsilon })\right\} _{\varepsilon \in (0,\varepsilon _0)}\) is precompact in the sense that for all sequences \(\varepsilon _n\rightarrow 0\) there exist a sub-sequence \(\varepsilon _{n_k}\rightarrow 0\) and \(w\in \mathcal H_0\) such that \(\Vert K_{\varepsilon _{n_k}}(u_{\varepsilon _{n_k}})-E_{\varepsilon _{n_k}}(w)\Vert _{\mathcal H_{\varepsilon _{n_k}}}\rightarrow 0\) as \(k\rightarrow +\infty \).

We also recall the following theorem, where by spectral convergence of a family of operators we mean the convergence of the eigenvalues and the convergence of the eigenfunctions in the sense of [33], see also [17, §2].

Theorem 3.6

Let \(\left\{ K_{\varepsilon }\right\} _{\varepsilon \in [0,\varepsilon _0)}\) be non-negative, compact self-adjoint operators in the Hilbert spaces \(\mathcal H_{\varepsilon }\). Assume that their eigenvalues are given by \(\left\{ \sigma _j(\varepsilon )\right\} _{j=1}^{\infty }\). If \(K_{\varepsilon }\) compactly converge to \(K_0\), then there is spectral convergence of \(K_{\varepsilon }\) to \(K_0\) as \(\varepsilon \rightarrow 0\). In particular, for every \(j\in \mathbb N\) \(\sigma _j(\varepsilon )\rightarrow \sigma _j(0)\,,\ \ \ \mathrm{as\ }\varepsilon \rightarrow 0\).

Let \(T_D:\mathcal {H}^2_{0,D}(\Omega )\rightarrow \mathcal {H}^2_{0,D}(\Omega )\) be defined by \(T_D=B_D^{(-1)}\circ J_1\), where \(B_D\) is the operator from \(\mathcal {H}^2_{0,D}(\Omega )\) to its dual \((\mathcal {H}^2_{0,D}(\Omega ))'\) given by

$$\begin{aligned} B_D(v)[\varphi ]=\mathcal Q_{\sigma }(v,\varphi )\,,\ \ \ \forall v,\varphi \in \mathcal {H}^2_{0,D}(\Omega ), \end{aligned}$$

and \(J_1\) is defined in (3.6). By the Riesz Theorem it follows that \(B_D\) is a surjective isometry. The operator \(T_D\) is the resolvent operator associated with problem (3.1) and plays the same role of \(T_{\mu ,D}\) defined in (3.7). In fact, as in the proof of Theorem 3.3 it is possible to show that \(T_D\) admits an increasing sequence of non-zero eigenvalues \(\left\{ q_j\right\} _{j=1}^{\infty }\) bounded from above and converging to 0. Moreover, a number \(q\ne 0\) is an eigenvalue of \(T_D\) if and only if \(\eta =\frac{1}{q}\) is an eigenvalue of (3.1), with the same eigenfunctions.

We have now a family of compact self-adjoint operators \(T_{\mu ,D}\) each defined on the Hilbert space \(H^2(\Omega )\) endowed with the scalar product \(\mathcal Q_{\mu ,D}\), and the compact self-adjoint operator \(T_D\) defined on \(\mathcal {H}^2_{0,D}(\Omega )\) endowed with the scalar product \(\mathcal Q_{\sigma }\). We are ready to state and prove the following theorem.

Theorem 3.7

The family of operators \(\left\{ T_{\mu ,D}\right\} _{\mu \in (-\infty ,0)}\) compactly converges to \(T_D\) as \(\mu \rightarrow -\infty \). In particular,

$$\begin{aligned} \lim _{\mu \rightarrow -\infty }\lambda _j(\mu )=\eta _j, \end{aligned}$$
(3.11)

for all \(j\in \mathbb N\), where \(\eta _j\) are the eigenvalues of (3.1).

Proof

For each \(\mu \in (-\infty ,0)\) we define the map \(E_{\mu }\equiv E:\mathcal {H}^2_{0,D}(\Omega )\rightarrow H^2(\Omega )\) simply by setting \(E(u)=u\), for all \(u\in \mathcal H^2_{0,D}(\Omega )\).

In view of Definition 3.5, we have to prove that

  1. i)

    if \(\left\{ u_{\mu }\right\} _{\mu <0}\subset H^2(\Omega )\) and \(u\in \mathcal {H}^2_{0,D}(\Omega )\) are such that \(\mathcal Q_{\mu ,D}(u_{\mu }-u,u_{\mu }-u)\rightarrow 0\) as \(\mu \rightarrow -\infty \), then

    $$\begin{aligned} \mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu })-T_D(u),T_{\mu ,D}(u_{\mu })-T_D(u))\rightarrow 0\,,\mathrm{\ \ \ as\ }\mu \rightarrow -\infty ; \end{aligned}$$
  2. ii)

    if \(\left\{ u_{\mu }\right\} _{\mu <0}\subset H^2(\Omega )\) is such that \(\mathcal Q_{\mu ,D}(u_{\mu },u_{\mu })=1\) for all \(\mu <0\), then for every sequence \(\mu _n\rightarrow -\infty \) there exists a sub-sequence \(\mu _{n_k}\rightarrow -\infty \) and \(v\in \mathcal H^2_{0,D}(\Omega )\) such that

    $$\begin{aligned} \mathcal Q_{{\mu _{n_k}},D}(T_{{\mu _{n_k}},D}(u_{{\mu _{n_k}}})-v,T_{{\mu _{n_k}},D}(u_{{\mu _{n_k}}})-v)\rightarrow 0\,,\mathrm{\ \ \ as\ }k\rightarrow +\infty . \end{aligned}$$
    (3.12)

We start by proving i). By the assumptions in i), it follows that \(u_{\mu }\) is uniformly bounded in \(H^2(\Omega )\) for \(\mu \) in a neighborhood of \(-\infty \). Indeed, by definition

$$\begin{aligned} \mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu }),\varphi )=\int _{\partial \Omega }\frac{\partial u_{\mu }}{\partial \nu }\frac{\partial \varphi }{\partial \nu }d\sigma \,,\ \ \ \forall \varphi \in H^2(\Omega ), \end{aligned}$$
(3.13)

hence, by choosing \(\varphi =T_{\mu ,D}(u_{\mu })\), we find that the family \(\left\{ T_{\mu ,D}(u_{\mu })\right\} _{\mu <0}\) is bounded in \(H^2(\Omega )\). Thus, possibly passing to a sub-sequence, \(T_{\mu ,D}(u_{\mu })\rightharpoonup v\) in \(H^2(\Omega )\), and \(\gamma _0(T_{\mu ,D}(u_{\mu }))\rightarrow \gamma _0(v)\) in \(L^2(\partial \Omega )\), as \(\mu \rightarrow -\infty \), which implies that \(\gamma _0(v)=0\) since the term \(-\mu \int _{\partial \Omega }T_{\mu ,D}(u_{\mu })^2d\sigma \) is bounded in \(\mu \). Thus \(v\in \mathcal {H}^2_{0,D}(\Omega )\).

Choosing \(\varphi \in \mathcal H^2_{0,D}(\Omega )\) and passing to the limit in (3.13) we have that

$$\begin{aligned} \mathcal Q_{\sigma }(v,\varphi )=\int _{\partial \Omega }\frac{\partial u}{\partial \nu }\frac{\partial \varphi }{\partial \nu }d\sigma \,,\ \ \ \forall \varphi \in \mathcal {H}^2_{0,D}(\Omega ), \end{aligned}$$

hence \(v=T_D(u)\). Thus \(T_{\mu ,D}(u_{\mu })\rightharpoonup T_D(u)\) in \(H^2(\Omega )\). Moreover, the convergence is stronger because

$$\begin{aligned}&\lim _{\mu \rightarrow -\infty }\mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu })-T_D(u),T_{\mu ,D}(u_{\mu })-T_D(u))\\&\quad =\lim _{\mu \rightarrow -\infty }\left( \mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu }),T_{\mu ,D}(u_{\mu }))\right. \\&\left. -2\mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu }),T_D(u))+\mathcal Q_{\mu ,D}(T_D(u),T_D(u))\right) \\&\quad =\mathcal Q_{\sigma }(T_D(u),T_D(u))-2\mathcal Q_{\sigma }(T_D(u),T_D(u))+\mathcal Q_{\sigma }(T_D(u),T_D(u))=0, \end{aligned}$$

which proves point i).

Note that the equality \(\lim _{\mu \rightarrow -\infty }\mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu }),T_{\mu ,D}(u_{\mu }))=\mathcal Q_{\sigma }(T_D(u),T_D(u))\) is a consequence of

$$\begin{aligned}&\lim _{\mu \rightarrow -\infty }\mathcal Q_{\mu ,D}(T_{\mu ,D}(u_{\mu }),T_{\mu ,D}(u_{\mu }))=\lim _{\mu \rightarrow -\infty }\int _{\partial \Omega }\frac{\partial u_{\mu }}{\partial \nu }\frac{\partial T_{\mu ,D}(u_{\mu })}{\partial \nu }d\sigma \\&\quad =\int _{\partial \Omega }\frac{\partial u}{\partial \nu }\frac{\partial T_D(u)}{\partial \nu }d\sigma =\mathcal Q_{\sigma }(T_D(u),T_D(u)). \end{aligned}$$

The proof of point ii) is similar. If \(\mathcal Q_{\mu ,D}(u_{\mu },u_{\mu })=1\), up to sub-sequences \(u_{\mu }\rightharpoonup u\in H^2(\Omega )\), \(\gamma _0(u_{\mu })\rightarrow \gamma _0(u)\), and \(\gamma _1(u_{\mu })\rightarrow \gamma _1(u)\) as \(\mu \rightarrow -\infty \). Moreover, \(\Vert \gamma _0(u_{\mu })\Vert _{L^2(\partial \Omega )}^2\le -\frac{1}{\mu }\), hence \(\Vert \gamma _0(u_{\mu })\Vert _{L^2(\partial \Omega )}^2\rightarrow 0\) as \(\mu \rightarrow -\infty \). This implies that \(\gamma _0(u)=0\) and that \(u\in \mathcal {H}^2_{0,D}(\Omega )\). Then it is possible to repeat the same arguments above to conclude the validity of (3.12) with \(v=T_D(u)\).

Thus \(T_{\mu ,D}\) compactly converges to \(T_D\) and (3.11) follows by Theorem 3.6. \(\square \)

Remark 3.8

We also note that each eigenvalue \(\lambda _j(\mu )\) is non-increasing with respect to \(\mu \), for \(\mu \in (-\infty ,0)\). In fact from the Min-Max Principle (3.8) it immediately follows that for all \(j\in \mathbb N\), \(\lambda _j(\mu _1)\le \lambda _j(\mu _2)\) if \(\mu _1>\mu _2\).

Now we consider the behavior of the first eigenvalue as \(\mu \rightarrow 0^-\).

Lemma 3.9

We have

$$\begin{aligned} \lim _{\mu \rightarrow 0^-}\lambda _1(\mu )=0 \end{aligned}$$

Proof

Let \(p\in \mathbb R^N\) be fixed. From (3.8) we get

$$\begin{aligned} 0<\lambda _1(\mu )=\min _{\begin{array}{c} v\in H^2(\Omega )\\ v\ne 0 \end{array}}\frac{\mathcal {Q}_{\mu ,D}(v,v)}{\int _{\partial \Omega }\left( \frac{\partial v}{\partial \nu }\right) ^2d\sigma }\le \frac{\mathcal {Q}_{\mu ,D}(p\cdot x,p\cdot x)}{\int _{\partial \Omega }(p\cdot \nu )^2d\sigma }=-\mu \frac{\int _{\partial \Omega }(p\cdot x)^2d\sigma }{\int _{\partial \Omega }(p\cdot \nu )^2d\sigma }, \end{aligned}$$

for all \(\mu \in (-\infty ,0)\). By letting \(\mu \rightarrow 0^-\) we obtain the result. \(\square \)

The \(BS_{\lambda }\) eigenvalue problem

The weak formulation of problem (\(\mathrm{BS}_{\lambda }\)) reads

$$\begin{aligned}&\int _{\Omega }(1-\sigma )D^2 u:D^2\varphi +\sigma \Delta u\Delta \varphi dx-\lambda \int _{\partial \Omega }\frac{\partial u}{\partial \nu }\frac{\partial \varphi }{\partial \nu }d\sigma \nonumber \\&=\mu (\lambda )\int _{\partial \Omega }u\varphi d\sigma \,,\ \ \ \forall \varphi \in H^2(\Omega ), \end{aligned}$$
(3.14)

in the unknowns \(u\in H^2(\Omega )\), \(\mu (\lambda )\in \mathbb R\), and can be re-written as

$$\begin{aligned} \mathcal {Q}_{\lambda ,N}(u,\varphi )=\mu (\lambda )\left( \gamma _0(u),\gamma _0(\varphi )\right) _{\partial \Omega }\,,\ \ \ \forall \varphi \in H^2(\Omega ). \end{aligned}$$

We prove that for all \(\lambda <\eta _1\), where \(\eta _1\) is the first eigenvalue of (\(\mathrm{DBS}\)), problem (\(\mathrm{BS}_{\lambda }\)) admits an increasing sequence of eigenvalues of finite multiplicity diverging to \(+\infty \) and the corresponding eigenfunctions form a basis of \(H^2_{\lambda ,N}(\Omega )\), where \(H^2_{\lambda ,N}(\Omega )\) denotes the orthogonal complement of \(\mathcal {H}^2_{0,D}(\Omega )\) in \(H^2(\Omega )\) with respect to \(\mathcal {Q}_{\lambda ,N}\), that is

$$\begin{aligned} H^2_{\lambda ,N}(\Omega )=\left\{ u\in H^2(\Omega ):\mathcal {Q}_{\lambda ,N}(u,\varphi )=0\,,\ \forall \varphi \in \mathcal {H}^2_{0,D}(\Omega )\right\} . \end{aligned}$$
(3.15)

Since in general \(\mathcal Q_{\lambda ,N}\) is not a scalar product, we find it convenient to consider on \(H^2(\Omega )\) the norm

$$\begin{aligned} \Vert u\Vert _{\lambda ,N}^2=\mathcal {Q}_{\lambda ,N}(u,u)+b\Vert \gamma _0(u)\Vert _{L^2(\Omega )}^2, \end{aligned}$$
(3.16)

where \(b>0\) is a fixed number which is chosen as follows. If \(\lambda <0\), no restrictions are required on \(b>0\), since the norm \(\Vert \cdot \Vert _{\lambda ,N}\) is equivalent to the standard norm of \(H^2(\Omega )\) for all \(b>0\). Assume now that \(0\le \lambda <\eta _1\). From Theorem 3.7 and Lemma 3.9 we have that \((0,\eta _1)\subseteq \lambda _1((-\infty ,0))\), hence there exists \(\mu \in (-\infty ,0)\) and \(\varepsilon \in (0,1)\) such that \(\lambda _1(\mu )=\frac{\lambda +\varepsilon }{1-\varepsilon }<\eta _1\). Then

$$\begin{aligned}&\mathcal Q_{\lambda ,N}(u,u)=\varepsilon \mathcal Q_{-1,N}(u,u)+(1-\varepsilon )\mathcal Q_{\lambda _1(\mu ),N}(u,u)\nonumber \\&\ge \varepsilon \mathcal Q_{-1,N}(u,u)+(1-\varepsilon )\mu \Vert \gamma _0(u)\Vert ^2_{L^2(\partial \Omega )}. \end{aligned}$$
(3.17)

Thus, by choosing any b satisfying

$$\begin{aligned} b>-(1-\varepsilon )\mu , \end{aligned}$$
(3.18)

it follows by (3.17) and (3.18) that \(\Vert \cdot \Vert _{\lambda ,N}\) is a norm equivalent to the standard norm of \(H^2(\Omega )\).

The norm \(\Vert \cdot \Vert _{\lambda ,N}\) is associated with the scalar product defined by

$$\begin{aligned} \langle u,\varphi \rangle _{\lambda ,N}=\mathcal {Q}_{\lambda ,N}(u,\varphi )+b(\gamma _0(u),\gamma _0(\varphi ))_{\partial \Omega }, \end{aligned}$$
(3.19)

for all \(u,\varphi \in H^2(\Omega )\).

We now recast problem (3.14) in the form of an eigenvalue problem for a compact self-adjoint operator acting on a Hilbert space. To do so, we define the operator \(B_{\lambda ,N}\) from \(H^2(\Omega )\) to its dual \((H^2(\Omega ))'\) by setting

$$\begin{aligned} B_{\lambda ,N}(u)[\varphi ]=\langle u,\varphi \rangle _{\lambda ,N}\,,\ \ \ \forall u,\varphi \in H^2(\Omega ). \end{aligned}$$

By the Riesz Theorem it follows that \(B_{\lambda ,N}\) is a surjective isometry. Then we consider the operator \(J_0\) from \(H^2(\Omega )\) to \((H^2(\Omega ))'\) defined by

$$\begin{aligned} J_0(u)[\varphi ]=(\gamma _0(u),\gamma _0(\varphi ))_{\partial \Omega }\,,\ \ \ \forall u,\varphi \in H^2(\Omega ). \end{aligned}$$
(3.20)

The operator \(J_0\) is compact since \(\gamma _0\) is a compact operator from \(H^2(\Omega )\) to \(L^2(\partial \Omega )\). Finally, we set

$$\begin{aligned} T_{\lambda ,N}=B_{\lambda ,N}^{(-1)}\circ J_0. \end{aligned}$$
(3.21)

From the compactness of \(J_0\) and the boundedness of \(B_{\lambda ,N}\) it follows that \(T_{\lambda ,N}\) is a compact operator from \(H^2(\Omega )\) to itself. Moreover, \( \langle B_{\lambda ,N}(u),\varphi \rangle _{\lambda ,N}=(\gamma _0(u),\gamma _0(\varphi ))_{\partial \Omega }\), for all \(u,\varphi \in H^2(\Omega )\), hence \(T_{\lambda ,N}\) is self-adjoint.

Note that \(\mathrm{Ker}\,T_{\lambda ,N}=\mathrm{Ker}\,J_0=\mathcal {H}^2_{0,D}(\Omega )\) and the non-zero eigenvalues of \(T_{\lambda ,N}\) coincide with the reciprocals of the eigenvalues of (3.14) shifted by b, the eigenfunctions being the same.

We are now ready to prove the following theorem.

Theorem 3.10

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\), \(\sigma \in \big (-\frac{1}{N-1},1\big )\), and \(\lambda <\eta _1\). Then the eigenvalues of (3.14) have finite multiplicity and are given by a non-decreasing sequence of real numbers \(\left\{ \mu _j(\lambda )\right\} _{j=1}^{\infty }\) defined by

$$\begin{aligned} \mu _j(\lambda )=\min _{\begin{array}{c} U\subset H^2(\Omega )\\ \mathrm{dim}U=j \end{array}}\max _{\begin{array}{c} u\in U\\ \gamma _0(u)\ne 0 \end{array}}\frac{\mathcal {Q}_{\lambda ,N}(u,u)}{\int _{\partial \Omega }u^2d\sigma }, \end{aligned}$$
(3.22)

where each eigenvalue is repeated according to its multiplicity. Moreover, there exists a Hilbert basis \(\left\{ u_{j,\lambda }\right\} _{j=1}^{\infty }\) of \(H^2_{\lambda ,N}(\Omega )\) (endowed with the scalar product (3.19)) of eigenfunctions \(u_{j,\lambda }\) associated with the eigenvalues \(\mu _j(\lambda )\) and the following statements hold:

  1. i)

    If \(\lambda <0\) then \(\mu _1(\lambda )=0\) is an eigenvalue of multiplicity one and the corresponding eigenfunctions are the constant functions. Moreover, if \(\tilde{u}_{j,\lambda }\) denote the normalizations of \(u_{j,\lambda }\) with respect to \(\mathcal {Q}_{\lambda ,N}\) for all \(j\ge 2\), the functions \(\hat{u}_{j,\lambda }:=\sqrt{\mu _{j}(\lambda )}\gamma _0(\tilde{u}_{j,\lambda })\), \(j\ge 2\), and \(\hat{u}_{1,\lambda }:=|\partial \Omega |^{-1/2}\) define a Hilbert basis of \(L^2(\partial \Omega )\) with respect to its standard scalar product.

  2. ii)

    If \(0\le \lambda <\eta _1\), then \(\mu (\lambda )=0\) is an eigenvalue. Moreover, if \(\mu _{j_0}(\lambda )\) is the first positive eigenvalue, and \(\tilde{u}_{j,\lambda }\) denote the normalizations of \(u_{j,\lambda }\) with respect to \(\mathcal {Q}_{\lambda ,N}\) for all \(j\ge j_0\), and \(\left\{ \hat{u}_{j,\lambda }\right\} _{j=1}^{j_0-1}\) denotes a orthonormal basis with respect to the \(L^2(\partial \Omega )\) scalar product of the eigenspace associated to \(\mu _1(\lambda ),...,\mu _{j_0-1}(\lambda )\) restricted to \(\partial \Omega \), then the functions \(\hat{u}_{j,\lambda }:=\sqrt{\mu _{j}(\lambda )}\gamma _0(\tilde{u}_{j,\lambda })\), \(j\ge j_0\), and \(\left\{ \hat{u}_{j,0}\right\} _{j=1}^{j_0-1}\), define a Hilbert basis of \(L^2(\partial \Omega )\) with respect to its standard scalar product. Finally, if \(\lambda =0\), then \(j_0=N+2\) and the eigenspace corresponding to \(\mu _1(0)=\cdots =\mu _{N+1}(0)=0\) is generated by \(\left\{ 1,x_1,...,x_N\right\} \); if \(\lambda >0\), then \(\mu _1(\lambda )<0\).

Proof

Since \(\mathrm{Ker}\,J_0=\mathcal {H}^2_{0,D}(\Omega )\), by the Hilbert-Schmidt Theorem applied to \(T_{\lambda ,N}\) it follows that \(T_{\lambda ,N}\) admits a non-increasing sequence of positive eigenvalues \(\left\{ p_j\right\} _{j=1}^{\infty }\) bounded from above, converging to zero and a corresponding Hilbert basis \(\left\{ u_{j,\lambda }\right\} \) of eigenfunctions of \(H^2_{\lambda ,N}(\Omega )\). We note that \(p\ne 0\) is an eigenvalue of \(T_{\lambda ,N}\) if and only if \(\mu =\frac{1}{p}-b\) is an eigenvalue of (3.14), the eigenfunction being the same.

Formula (3.22) follows from the standard min-max formula for the eigenvalues of compact self-adjoint operators.

If \(\lambda <0\), then \(\mu _1(\lambda )=0\) and a corresponding eigenfunction \(u_{1,\lambda }\) satisfies \(D^2u_{1,\lambda }=0\) in \(\Omega \), hence it is a linear function; moreover, since \(\frac{\partial u_{1,\lambda }}{\partial \nu }=0\) on \(\partial \Omega \), \(u_{1,\lambda }\) has to be constant.

If \(\lambda =0\), then \(\mu =0\) is an eigenvalue and a corresponding eigenfunction is a linear function. Hence \(\mu _1(0)=\cdots =\mu _{N+1}(0)=0\) and the associated eigenspace is spanned by \(\left\{ 1,x_1,...,x_N\right\} \).

If \(0<\lambda <\eta _1\), then by (3.11) and Lemma 3.9, there exists \(\mu <0\) such that \(\lambda _1(\mu )=\lambda \), hence \(\mu \) is an eigenvalue of (3.14). Moreover, by definition we have that for all \(u\in H^2(\Omega )\) with \(\gamma _0(u)\ne 0\)

$$\begin{aligned} \frac{Q_{\lambda ,N}(u,u)}{\int _{\partial \Omega }u^2d\sigma }=\frac{Q_{\lambda _1(\mu ),N}(u,u)}{\int _{\partial \Omega }u^2d\sigma }\ge \mu , \end{aligned}$$

hence \(\mu _1(\lambda )=\mu <0\).

To prove the final part of the theorem, we recast problem (3.14) into an eigenvalue problem for the compact self-adjoint operator \(T_{\lambda ,N}'=\gamma _0\circ B_{\lambda ,N}^{(-1)}\circ J_0'\), where \(J_0'\) denotes the map from \(L^2(\partial \Omega )\) to the dual of \(H^2(\Omega )\) defined by

$$\begin{aligned} J_0'(u)[\varphi ]=(u,\gamma _0(\varphi ))_{\partial \Omega }\,,\ \ \ \forall u\in L^2(\partial \Omega ),\varphi \in H^2(\Omega ). \end{aligned}$$

We apply again the Hilbert-Schmidt Theorem and observe that \(T_{\lambda ,N}\) and \(T_{\lambda ,N}'\) admit the same non-zero eigenvalues and that the eigenfunctions of \(T_{\lambda ,N}'\) are exactly the traces of the eigenfunctions of \(T_{\lambda ,N}\). From (3.14) we deduce that if we normalize the eigenfunction \(u_{j,\lambda }\) of \(T_{\lambda ,N}\) associated with positive eigenvalues and we denote them by \(\tilde{u}_{j,\lambda }\), then the normalization of their traces in \(L^2(\partial \Omega )\) are obtained by multiplying \(\gamma _0(\tilde{u}_{j,\lambda })\) by \(\sqrt{\mu _j(\lambda )}\). The rest of the proof easily follows. \(\square \)

As we have done for problem (3.4), we present now a few results on the behavior of the eigenvalues of (3.14) for \(\lambda \in (-\infty ,\eta _1)\). We have the following theorem on the Lipschitz continuity of eigenvalues, the proof of which is similar to that of Theorem 3.4 and is accordingly omitted.

Theorem 3.11

For any \(j\in \mathbb N\) and \(\delta >0\), the functions \(\mu _j:(-\infty ,\eta _1-\delta ]\rightarrow [0,+\infty )\) which takes \(\lambda \in (-\infty ,\eta _1-\delta ]\) to \(\mu _j(\lambda )\in \mathbb R\) are Lipschitz continuous on \((-\infty ,\eta _1-\delta ]\).

We now investigate the behavior of the eigenvalues \(\mu _j(\lambda )\) as \(\lambda \rightarrow -\infty \). In order state the analogue of Theorem 3.7, we consider the operator \(T_N:\mathcal {H}^2_{0,N}(\Omega )\rightarrow \mathcal {H}^2_{0,N}(\Omega )\) defined by \(T_N=B_N^{(-1)}\circ J_0\), where \(B_N\) is the operator from \(\mathcal {H}^2_{0,N}(\Omega )\) to its dual \((\mathcal {H}^2_{0,N}(\Omega ))'\) given by

$$\begin{aligned} B_N(v)[\varphi ]=\mathcal Q_{\sigma }(v,\varphi )+b(\gamma _0(v),\gamma _0(\varphi ))_{\partial \Omega }\,,\ \ \ \forall v,\varphi \in \mathcal {H}^2_{0,N}(\Omega ), \end{aligned}$$
(3.23)

and b has the same value as in the definition of the operator \(T_{\lambda ,N}\), see (3.18), and \(J_0\) is defined in (3.20). Note that the constant b can be chosen to be independent of \(\lambda \) for \(\lambda <0\). By the Riesz Theorem it follows that \(B_N\) is a surjective isometry. The operator \(T_N\) is the resolvent operator associated with problem (3.3) and plays the same role of \(T_{\lambda ,N}\) defined in (3.21). In fact, as in the proof of Theorem 3.10 it is possible to show that \(T_N\) admits an increasing sequence of non-zero eigenvalues \(\left\{ p_j\right\} _{j=1}^{\infty }\) bounded from above and converging to 0. Moreover, a number \(p\ne 0\) is an eigenvalue of \(T_N\) if and only if \(\xi =\frac{1}{p}-b\) is an eigenvalue of (3.3), with the same eigenfunctions.

We have now a family of compact self-adjoint operators \(T_{\lambda ,N}\) each defined on the Hilbert space \(H^2(\Omega )\) endowed with the scalar product (3.19), and the compact self-adjoint operator \(T_N\) defined on \(\mathcal {H}^2_{0,N}(\Omega )\) endowed with the scalar product defined by the right-hand side of (3.23). We have the following theorem, the proof of which is similar to that of Theorem 3.7 and is accordingly omitted.

Theorem 3.12

The family of operators \(\{T_{\lambda ,N}\}_{\lambda \in (-\infty ,\eta _1)}\) compactly converges to \(T_N\) as \(\lambda \rightarrow -\infty \). In particular,

$$\begin{aligned} \lim _{\lambda \rightarrow -\infty }\mu _j(\lambda )=\xi _j, \end{aligned}$$
(3.24)

for all \(j\in \mathbb N\), where \(\xi _j\) are the eigenvalues of (3.3).

Remark 3.13

We also note that each eigenvalue \(\mu _j(\lambda )\) is non-increasing with respect to \(\lambda \), for \(\lambda \in (-\infty ,\eta _1)\). In fact from the Min-Max Principle (3.22) it immediately follows that for all \(j\in \mathbb N\), \(\mu _j(\lambda _1)\le \mu _j(\lambda _2)\) if \(\lambda _1>\lambda _2\).

Characterization of trace spaces of \(H^2(\Omega )\) via biharmonic Steklov eigenvalues

In this section we shall use the Hilbert basis of eigenfunctions \(v_{j,\mu }\) and \(\hat{v}_{j,\mu }\) given by Theorem 3.3 and the Hilbert basis of eigenfunctions \(u_{j,\lambda },\hat{u}_{j,\lambda }\) given by Theorem 3.10, for all \(\mu \in (-\infty ,0)\) and \(\lambda \in (-\infty ,\eta _1)\). We recall that by definition, the functions \(v_{j,\mu }\) and \(u_{j,\lambda }\) are normalized with respect to \(Q_{\mu ,D}(\cdot ,\cdot )\) and \(Q_{\lambda ,N}(\cdot ,\cdot )+b(\gamma _0(\cdot ),\gamma _0(\cdot ))_{\partial \Omega }\) respectively, while \(\hat{v}_{j,\mu }\) and \(\hat{u}_{j,\lambda }\) are normalized with respect to the standard scalar product of \(L^2(\partial \Omega )\).

We will also denote by \(l^2\) the space of sequences \(s=(s_j)_{j=1}^{\infty }\) of real numbers satisfying \(\Vert s\Vert _{l^2}^2=\sum _{j=1}^{\infty }s_j^2<\infty \).

We define the spaces

$$\begin{aligned} \mathcal S^{\frac{3}{2}}(\partial \Omega )=\mathcal S^{\frac{3}{2}}_{\lambda }(\partial \Omega ):=\left\{ f\in L^2(\partial \Omega ): f=\sum _{j=1}^{\infty }\hat{a}_j\hat{u}_{j,\lambda }\mathrm{\ with\ }\left( \sqrt{|\mu _j(\lambda )|}\hat{a}_j\right) _{j=1}^{\infty }\in l^2\right\} , \end{aligned}$$
(4.1)

and

$$\begin{aligned} \mathcal S^{\frac{1}{2}}(\partial \Omega )=\mathcal S^{\frac{1}{2}}_{\mu }(\partial \Omega ):=\left\{ f\in L^2(\partial \Omega ): f=\sum _{j=1}^{\infty }\hat{b}_j\hat{v}_{j,\mu }\mathrm{\ with\ }\left( \sqrt{\lambda _j(\mu )}\hat{b}_j\right) _{j=1}^{\infty }\in l^2\right\} . \end{aligned}$$
(4.2)

These spaces are endowed with the natural norms defined by

$$\begin{aligned} \Vert f\Vert _{\mathcal S^{\frac{3}{2}}_{\lambda }(\partial \Omega )}^2=\sum _{j=1}^{j_0-1}\hat{a}_j^2+\sum _{j=j_{0}}^{\infty }\mu _j(\lambda )\hat{a}_j^2, \end{aligned}$$

where \(j_0\) is as in Theorem 3.10, and

$$\begin{aligned} \Vert f\Vert _{\mathcal S^{\frac{1}{2}}_{\mu }(\partial \Omega )}^2=\sum _{j=1}^{\infty }\lambda _j(\mu )\hat{b}_j^2. \end{aligned}$$

Recall that if \(\lambda =0\), \(j_0=N+2\) and if \(\lambda <0\), then \(j_0=2\).

These spaces allow to describe the trace spaces for \(H^2(\Omega )\). In particular, \(\mathcal S^{\frac{3}{2}}_{\lambda }(\partial \Omega )\) and \(\mathcal S^{\frac{1}{2}}_{\mu }(\partial \Omega )\) turn out to be independent of \(\lambda \) and \(\mu \). Namely, we have the following.

Theorem 4.1

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\). Then

$$\begin{aligned} \gamma _0(H^2(\Omega ))=\gamma _0(H^2_{\lambda ,N}(\Omega ))=\mathcal S^{\frac{3}{2}}(\partial \Omega )\ (=\mathcal S^{\frac{3}{2}}_{\lambda }(\partial \Omega )) \end{aligned}$$
(4.3)

and

$$\begin{aligned} \gamma _1(H^2(\Omega ))=\gamma _1(H^2_{\mu ,D}(\Omega ))=\mathcal S^{\frac{1}{2}}(\partial \Omega )\ (=\mathcal S^{\frac{1}{2}}_{\mu }(\partial \Omega )). \end{aligned}$$
(4.4)

In particular, the spaces \(\mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\) and \(\mathcal S_{\mu }^{\frac{1}{2}}(\partial \Omega )\) do not depend on \(\lambda \in (-\infty ,\eta _1)\) and \(\mu \in (-\infty ,0)\).

Moreover, if \(\Omega \) is of class \(C^{2,1}\) then

$$\begin{aligned} \Gamma (H^2(\Omega ))=\mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$

hence

$$\begin{aligned} \mathcal S^{\frac{3}{2}}(\partial \Omega )=H^{\frac{3}{2}}(\partial \Omega ) \end{aligned}$$

and

$$\begin{aligned} \mathcal S^{\frac{1}{2}}(\partial \Omega )=H^{\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

Proof

Let us begin by proving (4.3). By the definition of \(H^2_{\lambda ,N}(\Omega )\) given in (3.15) and by Theorem 3.10 we have that any \(u\in H^2(\Omega )\) can be written as

$$\begin{aligned} u=u_{\lambda }+v_D \end{aligned}$$

where \(v_D\in \mathcal {H}^2_{0,D}(\Omega )\) and

$$\begin{aligned} u_{\lambda }=\sum _{j=1}^{\infty } a_j u_{j,\lambda } \end{aligned}$$

for some coefficients \(a_j\) satisfying \(\sum _{j=1}^{\infty }a_j^2<\infty \). Here \(\left\{ u_{j,\lambda }\right\} _{j=1}^{\infty }\) is a orthonormal basis of \(H^2_{\lambda ,N}(\Omega )\) with respect to the scalar product (3.19) with b satisfying (3.18). Let \(j_0\) be as in Theorem 3.10. Hence we can write

$$\begin{aligned} u_{\lambda }= & {} \sum _{j=1}^{j_0-1} a_j u_{j,\lambda }+\sum _{j=j_0}^{\infty } a_j u_{j,\lambda }\\= & {} \sum _{j=1}^{j_0-1} a_j u_{j,\lambda }+\sum _{j=j_0}^{\infty } \left( a_j \sqrt{\mathcal Q_{\lambda ,N}(u_{j,\lambda },u_{j,\lambda })}\right) \cdot \frac{u_{j,\lambda }}{\sqrt{\mathcal Q_{\lambda ,N}(u_{j,\lambda },u_{j,\lambda })}}\\= & {} \sum _{j=1}^{j_0-1} a_j u_{j,\lambda }+\sum _{j=j_0}^{\infty } \tilde{a}_j \tilde{u}_{j,\lambda }, \end{aligned}$$

where \(\tilde{u}_{j,\lambda }=\frac{u_{j,\lambda }}{\sqrt{\mathcal Q_{\lambda ,N}(u_{j,\lambda },u_{j,\lambda })}}\) are the eigenfunctions normalized with respect to \(\mathcal Q_{\lambda ,N}\) and \(\tilde{a}_j\) still satisfy \(\sum _{j=j_0+1}^{\infty }\tilde{a}_j^2 <\infty \) (in fact \(0<\mathcal Q_{\lambda ,N}(u_{j,\lambda },u_{j,\lambda })\le 1\) for all \(j\ge j_0\)).

Clearly \(\gamma _0(u)=\gamma _0(u_\lambda )\), hence by the continuity of the trace operator we have that

$$\begin{aligned} \gamma _0(u_{\lambda })= & {} \sum _{j=1}^{j_0-1} a_j \gamma _0 (u_{j,\lambda })+\sum _{j=j_0}^{\infty } \tilde{a}_j \gamma _0(\tilde{u}_{j,\lambda })\\= & {} \sum _{j=1}^{j_0-1} \frac{a_j}{\sqrt{\mu _j(\lambda )+b}}\cdot \left( \sqrt{\mu _j(\lambda )+b}\cdot \gamma _0 (u_{j,\lambda })\right) +\sum _{j=j_0}^{\infty } \frac{\tilde{a}_j}{\sqrt{\mu _j(\lambda )}}\cdot \gamma _0\left( \sqrt{\mu _j(\lambda )}\tilde{u}_{j,\lambda }\right) \\= & {} \sum _{j=1}^{j_0-1}\hat{a}_j\hat{u}_{j,\lambda }+\sum _{j=j_0}^{\infty }\hat{a}_j\hat{u}_{j,\lambda }=\sum _{j=1}^{\infty }\hat{a}_j\hat{u}_{j,\lambda }, \end{aligned}$$

where we have set

$$\begin{aligned} \hat{a}_j=\frac{a_j}{\sqrt{\mu _j(\lambda )+b}}\,,\ \ \ \hat{u}_{j,\lambda }=\sqrt{\mu _j(\lambda )+b}\cdot \gamma _0 (u_{j,\lambda }) \end{aligned}$$

for \(j=1,...,j_0-1\) and

$$\begin{aligned} \hat{a}_j=\frac{\tilde{a}_j}{\sqrt{\mu _j(\lambda )}}\,,\ \ \ \hat{u}_{j,\lambda }=\sqrt{\mu _j(\lambda )}\cdot \gamma _0 (\tilde{u}_{j,\lambda }) \end{aligned}$$

for \(j\ge j_0\). This proves that \(\gamma _0(H^2_{\lambda ,N}(\Omega ))\subseteq \mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\).

We prove now the opposite inclusion. Let \(f\in \mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\). Then \(f=\sum _{j=1}^{\infty }\hat{a}_j\hat{u}_{j,\lambda }\) with \(\sum _{j=1}^{\infty }|\mu _j(\lambda )|\hat{a}_j^2<\infty \). Let \(u:=\sum _{j=1}^{\infty }a_j u_{j,\lambda }\) where

$$\begin{aligned} a_j=\sqrt{\mu _j(\lambda )+b}\cdot \hat{a}_j \end{aligned}$$
(4.5)

By definition, \(u\in H^2(\Omega )\) since \(\sum _{j=1}^{\infty }a_j^2<\infty \). Moreover, we note that

$$\begin{aligned} f=\sum _{j=1}^{\infty }\hat{a}_j\hat{u}_{j,\lambda }=\sum _{j=1}^{\infty }\hat{a}_j\sqrt{\mu _j(\lambda )+b}\cdot \frac{\hat{u}_{j,\lambda }}{\sqrt{\mu _j(\lambda )+b}}=\sum _{j=1}^{\infty }\hat{a}_j\sqrt{\mu _j(\lambda )+b} \cdot \gamma _0(u_{j,\lambda }), \end{aligned}$$
(4.6)

hence \(f=\gamma _0(u)\in \gamma _0(H^2(\Omega ))\).

The proof of (4.4) follows the same lines as that of (4.3) and is accordingly omitted.

We deduce then that the spaces \(\mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\) and \(\mathcal S_{\mu }^{\frac{1}{2}}(\partial \Omega )\) do not depend on the particular choice of \(\lambda \in (-\infty ,\eta _1)\) and \(\mu \in (-\infty ,0)\). In particular, we have proved that \(\Gamma (H^2(\Omega ))\subseteq \mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\times \mathcal S_{\mu }^{\frac{1}{2}}(\partial \Omega )\).

Assume now that \(\Omega \) is of class \(C^{2,1}\). We prove that \(\mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\times \mathcal S_{\mu }^{\frac{1}{2}}(\partial \Omega )\subseteq \Gamma (H^2(\Omega ))\). This will imply \(\Gamma (H^2(\Omega ))=\mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\times \mathcal S_{\mu }^{\frac{1}{2}}(\partial \Omega )\).

Let \((f,g)\in \mathcal S_{\lambda }^{\frac{3}{2}}(\partial \Omega )\times \mathcal S_{\mu }^{\frac{1}{2}}(\partial \Omega )\). This means that \(f=\gamma _0(u_{\lambda })\), \(g=\gamma _1(v_{\mu })\) for some \(u_{\lambda }\in H^2_{\lambda ,N}(\Omega )\), \(v_{\mu }\in H^2_{\mu ,D}(\Omega )\). We claim that there exist \(v_D\in \mathcal {H}^2_{0,D}(\Omega )\) and \(u_N\in \mathcal {H}^2_{0,N}(\Omega )\) such that \(u_{\lambda }+v_D=v_{\mu }+u_N\). To do so, it suffices to prove the existence of \(v_D\in \mathcal H^2_{0,D}(\Omega )\) and \(u_N\in \mathcal H^2_{0,N}(\Omega )\) such that \(u_{\lambda }-v_{\mu }=u_N-v_D\). We claim that

$$\begin{aligned} H^2(\Omega )=\mathcal H^2_{0,D}(\Omega )+\mathcal H^2_{0,N}(\Omega ). \end{aligned}$$
(4.7)

Indeed, given \(u\in H^2(\Omega )\), one can find by the classical Total Trace Theorem a function \(u_1\in H^2(\Omega )\) such that \(\gamma _0(u_1)=0\) and \(\gamma _1(u_1)=\gamma _1(u)\). Thus \(u=u_1+(u-u_1)\) with \(\gamma _1(u-u_1)=0\) and the claim is proved. Thus the existence of functions \(v_D\) and \(u_N\) follows by (4.7) and the function \(u=u_{\lambda }+v_D=v_{\mu }+u_N\) is such that \(f=\gamma _0(u)\) and \(g=\gamma _1(u)\). \(\square \)

Remark 4.2

Theorem 4.1 gives an explicit spectral characterization of the space \(\gamma _0(H^2(\Omega ))\) of traces of functions in \(H^2(\Omega )\) when \(\Omega \) is a bounded domain of class \(C^{0,1}\) in \(\mathbb R^N\). This space corresponds to \(H^{\frac{3}{2}}(\partial \Omega )\) when \(\Omega \) is of class \(C^{2,1}\). In this case explicit descriptions of the space \(H^\frac{3}{2}(\partial \Omega )\) are available in the literature and typically are given by means local charts and explicit representation of derivatives, see e.g., [22, 31].

For domains of class \(C^{0,1}\), it is not clear what is the appropriate definition of \(H^{\frac{3}{2}}(\partial \Omega )\). Sometimes \(H^{\frac{3}{2}}(\partial \Omega )\) is defined just by setting

$$\begin{aligned} H^{\frac{3}{2}}(\partial \Omega ):=\gamma _0(H^2(\Omega )). \end{aligned}$$

According to this definition, Theorem 4.1 implies that \(H^{\frac{3}{2}}(\partial \Omega )=\mathcal S^{\frac{3}{2}}(\partial \Omega )\) also for domains of class \(C^{0,1}\).

From Theorem 4.1 it follows that if \(\Omega \) is a domain of class \(C^{0,1}\), then

$$\begin{aligned} \Gamma (H^2(\Omega ))\subseteq \mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$
(4.8)

and equality holds if \(\Omega \) is of class \(C^{2,1}\). We observe that if \(\Omega \) is not of class \(C^{2,1}\), then in general equality does not hold in (4.8). Indeed, we have the following counterexample.

Counterexample 4.3

Let \(\Omega =(0,1)\times (0,1)\) be unit square in \(\mathbb R^2\). We prove that

$$\begin{aligned} \Gamma (H^2(\Omega ))\subsetneq \mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

To do so, we consider the real-valued function \(\varphi \) defined in \(\Omega \) by \(\varphi (x_1,x_2)=x_1\) for all \((x_1,x_2)\in \Omega \) and we prove that the couple \((\gamma _0(\varphi ),0)\in (\mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega ))\setminus \Gamma (H^2(\Omega ))\). It is obvious that \(\gamma _0(\varphi )\in \mathcal S^{\frac{3}{2}}(\partial \Omega )\) since \(\varphi \in H^2(\Omega )\). Assume now by contradiction that \((\gamma _0(\varphi ),0)\in \Gamma (H^2(\Omega ))\), that is, there exists \(u\in H^2(\Omega )\) such that \(\gamma _0(u)=\gamma _0(\varphi )\) and \(\gamma _1(u)=0\). Clearly, since \(\gamma _0(u)=\gamma _0(\varphi )\), there exists \(v_D\in \mathcal {H}^2_{0,D}\) such that

$$\begin{aligned} u=\varphi +v_D \end{aligned}$$

and hence

$$\begin{aligned} \gamma _1(v_D)=\gamma _1(u)-\gamma _1(\varphi )=-\nabla x_1\cdot \nu _{|_{\partial \Omega }}=-\nu _1. \end{aligned}$$

It follows that \(v_D\) is a function in \(H^2(\Omega )\) such that \(\gamma _0(v_D)=0\) and \(\gamma _1(v_D)=-\nu _1\), but this opposes a well-known necessary (and sufficient) condition for a couple \((f,g)\in H^1(\partial \Omega )\times L^2(\partial \Omega )\) to belong to \(\Gamma (H^2(\Omega ))\), namely

$$\begin{aligned} \frac{\partial f}{\partial \tau }\tau +g\nu \in H^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$
(4.9)

where \(\tau \) is the unit tangent vector (positively oriented with respect to the outer unit \(\nu \) to \(\Omega \)), see [21, 22]. Indeed, the couple \((0,-\nu _1)\) does not satisfy condition (4.9).

In order to characterize those couples \((f,g)\in \mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega )\) which belong to \(\Gamma (H^2(\Omega ))\) when \(\Omega \) is of class \(C^{0,1}\), we need the spaces \(\mathscr {S}^{\frac{3}{2}}(\partial \Omega )\) and \(\mathscr {S}^{\frac{1}{2}}(\partial \Omega )\) defined by

$$\begin{aligned} \mathscr {S}^{\frac{3}{2}}(\partial \Omega ):=\gamma _0(\mathcal {H}^2_{0,N})=\gamma _0(\mathcal B_N(\Omega )) \end{aligned}$$

and

$$\begin{aligned} \mathscr {S}^{\frac{1}{2}}(\partial \Omega ):=\gamma _1(\mathcal {H}^2_{0,D})=\gamma _1(\mathcal B_D(\Omega )). \end{aligned}$$

The spaces \(\mathscr {S}^{\frac{3}{2}}(\partial \Omega )\) and \(\mathscr {S}^{\frac{1}{2}}(\partial \Omega )\) have explicit descriptions similar to those of \(\mathcal S^{\frac{3}{2}}_{\lambda }(\partial \Omega )\) and \(\mathcal S^{\frac{1}{2}}_{\mu }(\partial \Omega )\), namely

$$\begin{aligned} \mathscr {S} ^{\frac{3}{2}}(\partial \Omega )=\biggl \{ f\in L^2(\partial \Omega ):\ f=\sum _{j=1}^{\infty }\hat{c}_j\hat{u}_j\ \mathrm{with}\ ( \sqrt{\xi _j}\hat{c}_j)_{j=1}^{\infty }\in l^2\biggr \}\, . \end{aligned}$$
(4.10)

and

$$\begin{aligned} \mathscr {S} ^{\frac{1}{2}}(\partial \Omega )=\biggl \{ g\in L^2(\partial \Omega ):\ g=\sum _{j=1}^{\infty }\hat{d}_j\hat{v}_j\ \mathrm{with}\ (\sqrt{\eta _j}\hat{d}_j)_{j=1}^{\infty }\in l^2 \biggr \}\, . \end{aligned}$$
(4.11)

Here \(\hat{u}_j=\sqrt{\xi _j}\gamma _0(u_j)\), \(j\ge 2\), where \(\left\{ u_j\right\} _{j=1}^{\infty }\) is a Hilbert basis of eigenfunctions of problem (3.3), normalized with respect to \(\mathcal Q_{\sigma }\), with the understanding that \(u_1\) and \(\hat{u}_1\) equal the constant \(|\partial \Omega |^{-1/2}\), and \(\hat{v}_j=\sqrt{\eta _j}\gamma _1(v_j)\), where \(\left\{ v_j\right\} _{j=1}^{\infty }\) is a Hilbert basis of eigenfunctions of problem (3.1) normalized with respect to \(\mathcal Q_{\sigma }\).

Note that

$$\begin{aligned} \mathscr {S}^{\frac{3}{2}}(\partial \Omega )\times \mathscr {S}^{\frac{1}{2}}(\partial \Omega )=\Gamma (\mathcal H^2_{0,N}(\Omega )+\mathcal H^2_{0,D}(\Omega ))\subseteq \Gamma (H^2(\Omega )). \end{aligned}$$

One can see by similar arguments as in Counterexample 4.3 that in general \(\mathscr {S}^{\frac{3}{2}}(\partial \Omega )\times \mathscr {S}^{\frac{1}{2}}(\partial \Omega ) \subsetneq \Gamma (H^2(\Omega ))\) if \(\Omega \) is not of class \(C^{2,1}\), while equality occurs if \(\Omega \) is of class \(C^{2,1}\) by (4.7).

We are now ready to characterize the trace space \(\Gamma (H^2(\Omega ))\) for domains \(\Omega \) of class \(C^{0,1}\).

Theorem 4.4

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\). Let \((f,g)\in \mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega )= \mathcal S^{\frac{3}{2}}_{\lambda }(\partial \Omega )\times \mathcal S^{\frac{1}{2}}_{\mu }(\partial \Omega )\) be given by

$$\begin{aligned} f=\sum _{j=1}^{\infty }\hat{a}_j \hat{u}_{j,\lambda }\,,\ \ \ g=\sum _{j=1}^{\infty }\hat{b}_j \hat{v}_{j,\mu } \end{aligned}$$
(4.12)

for some \(\lambda \in (-\infty ,\eta _1)\), \(\mu \in (-\infty ,0)\), with \(\left( \sqrt{|\mu _j(\lambda )|}\hat{a}_j\right) _{j=1}^{\infty },\left( \sqrt{\lambda _j(\mu )}\hat{b}_j\right) _{j=1}^{\infty }\in l^2\). Then (fg) belongs to \(\Gamma (H^2(\Omega ))\) if and only if

$$\begin{aligned} \sum _{j=1}^{\infty }a_j\gamma _1(u_{j,\lambda })-g\in \mathscr {S}^{\frac{1}{2}}(\partial \Omega ), \end{aligned}$$
(4.13)

where \(a_j\) are given by (4.5).

Equivalently, (fg) belongs to \(\Gamma (H^2(\Omega ))\) if and only if

$$\begin{aligned} \sum _{j=1}^{\infty }b_j\gamma _0(v_{j,\mu })-f\in \mathscr {S}^{\frac{3}{2}}(\partial \Omega ), \end{aligned}$$
(4.14)

where \(b_j=\sqrt{\lambda _j(\mu )}\hat{b}_j\).

Proof

Assume that \((f,g)\in \Gamma (H^2(\Omega ))\). Then \(f=\gamma _0(u_{\lambda }+v_D)\) where \(v_D\in \mathcal {H}^2_{0,D}(\Omega )\) and \(u_{\lambda }=\sum _{j=1}^{\infty }a_ju_{j,\lambda }\) with the coefficients \(a_j\) given by (4.5). Moreover, \(g=\gamma _1(u_{\lambda }+v_D)\) by the continuity of the trace operator. We deduce that

$$\begin{aligned} \gamma _1(u_{\lambda })-g=-\gamma _1(v_D)\in \mathscr {S}^{\frac{1}{2}}(\partial \Omega ). \end{aligned}$$

This proves (4.13). Vice versa, assume that (4.13) holds. Then there exist \(v_D\in \mathcal {H}^2_{0,D}(\Omega )\) such that \(\gamma _1(v_D)=\sum _{j=1}^{\infty }a_j\gamma _1(u_{j,\lambda })-g\). Thus

$$\begin{aligned} \gamma _1\left( \sum _{j=1}^{\infty }a_ju_{j,\lambda }-v_D\right) =g \end{aligned}$$

and

$$\begin{aligned} \gamma _0\left( \sum _{j=1}^{\infty }a_ju_{j,\lambda }-v_D\right) =f \end{aligned}$$

by (4.6). The proof of the second part of the statement follows the same lines as that of the first part and is accordingly omitted. \(\square \)

Representation of the solutions to the Dirichlet problem

Using the Steklov expansions in (4.1) and (4.2) and the characterization of the total trace space \(\Gamma (H^2(\Omega ))\) given by Theorem 4.4 we are able to describe the solutions to the Dirichlet problem (1.4).

Corollary 4.5

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\), \((f,g)\in L^2(\partial \Omega )\times L^2(\partial \Omega )\). Then, there exists a solution \(u\in H^2(\Omega )\) to problem (1.4) if and only if the couple (fg) belongs to \(\mathcal S^{\frac{3}{2}}(\partial \Omega )\times \mathcal S^{\frac{1}{2}}(\partial \Omega )\) and satisfies condition (4.13) or, equivalently, condition (4.14). In this case, if fg are represented as in (4.12), then the solution u can be represented in each of the following two forms:

  1. i)

    if \(u_{\lambda }:=\sum _{j=1}^{\infty }a_j u_{j,\lambda }\) where \(a_j\) are given by (4.5) and \(g-\gamma _1(u_{\lambda })\) is represented by \(\sum _{j=1}^{\infty }\hat{d}_j\hat{v}_j\in \mathscr {S}^{\frac{1}{2}}(\partial \Omega )\), then

    $$\begin{aligned} u=u_{\lambda }+v_D \end{aligned}$$

    with \(u_{\lambda }\in H^2_{\lambda ,N}(\Omega )\) and \(v_D=\sum _{j=1}^{\infty }d_jv_j\in \mathcal B_D(\Omega )\), \(d_j=\sqrt{\eta _j}\hat{d}_j\) for all \(j\in \mathbb N\).

  2. ii)

    if \(v_{\mu }:=\sum _{j=1}^{\infty }b_j v_{j,\mu }\) where \(b_j=\sqrt{\lambda _j(\mu )}\hat{b}_j\) and \(f-\gamma _0(v_{\mu })\) is represented by \(\sum _{j=1}^{\infty }\hat{c}_j\hat{u}_j\in \mathscr {S}^{\frac{3}{2}}(\partial \Omega )\), then

    $$\begin{aligned} u=v_{\mu }+u_N \end{aligned}$$

    with \(v_{\mu }\in H^2_{\mu ,D}(\Omega )\) and \(u_N=\sum _{j=1}^{\infty }c_ju_j\in \mathcal B_N(\Omega )\), \(c_j=\sqrt{\xi _j}\hat{c}_j\) for all \(j\in \mathbb N\), \(j\ge 2\), \(c_1=\hat{c}_1\).

Moreover the solution u is unique

Proof

The first part of the statement is an immediate consequence of Theorem 4.4. Indeed, if there exists a solution \(u\in H^2(\Omega )\) then (fg) belongs to \(\Gamma (H^2(\Omega ))\), hence it satisfies (4.13) and (4.14).

Vice versa, if (fg) satisfies (4.13), then \(u_{\lambda }\in H^2_{\lambda ,N}(\Omega )\), \(v_D\in \mathcal B_D(\Omega )\) are well-defined and \(u=u_{\lambda }+v_D\) is a biharmonic function in \(H^2(\Omega )\) such that \(\gamma _0(u)=\gamma _0(u_{\lambda })=f\) and \(\gamma _1(v_D)=g-\gamma _1(u_{\lambda })\), hence \(\gamma _1(u)=\gamma _1(u_{\lambda }+v_D)=g\).

Similarly, if (fg) satisfies (4.14), then \(v_{\mu }\in H^2_{\mu ,D}(\Omega )\), \(u_N\in \mathcal B_N(\Omega )\) are well-defined and \(u=v_{\mu }+u_N\) is a biharmonic function in \(H^2(\Omega )\) such that \(\gamma _1(u)=\gamma _1(v_{\mu })=g\) and \(\gamma _0(u_N)=f-\gamma _0(v_{\mu })\), hence \(\gamma _0(u)=\gamma _0(v_{\mu }+u_N)=f\).

The uniqueness of the solution in \(H^2(\Omega )\) follows from the fact that a solution u in \(H^2(\Omega )\) of (1.4) with \(f=g=0\) must belong to \(H^2_0(\Omega )\) and, since it is biharmonic, it must also belong to the orthogonal of \(H^2_0(\Omega )\), hence \(u=0\). \(\square \)

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Correspondence to Pier Domenico Lamberti.

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The authors are very thankful to Professors Giles Auchmuty and Victor I. Burenkov for useful discussions and references. This work was partially supported by the research project “Progetto GNAMPA 2019—Analisi spettrale per operatori ellittici con condizioni di Steklov o parzialmente incernierate” and the research project BIRD191739/19 “Sensitivity analysis of partial differential equations in the mathematical theory of electromagnetism” of the University of Padova. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Appendices

Eigenvalues and eigenfunctions on the ball

In this section we compute the eigenvalues and the eigenfunctions of (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) when \(\Omega =B\) is the unit ball in \(\mathbb R^N\) centered at the origin and \(\sigma =0\). It is convenient to use spherical coordinates \((r,\theta )\), where \(\theta =(\theta _1,...,\theta _{N-1})\). The corresponding transformation of coordinates is

$$\begin{aligned} x_1= & {} r \cos (\theta _1),\\ x_2= & {} r\sin (\theta _1)\cos (\theta _2),\\ \vdots \\ x_{N-1}= & {} r\sin (\theta _1)\sin (\theta _2)\cdots \sin (\theta _{N-2})\cos (\theta _{N-1}),\\ x_N= & {} r\sin (\theta _1)\sin (\theta _2)\cdots \sin (\theta _{N-2})\sin (\theta _{N-1}), \end{aligned}$$

with \(\theta _1,...,\theta _{N-2}\in [0,\pi ]\), \(\theta _{N-1}\in [0,2\pi )\) (here it is understood that \(\theta _1\in [0,2\pi )\) if \(N=2\)).

The boundary conditions for fixed parameters \(\lambda ,\mu \in \mathbb R\)

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial ^2 u}{\partial \nu ^2}=\lambda \frac{\partial u}{\partial \nu }, &{} \mathrm{on\ }\partial B,\\ -\mathrm{div}_{\partial \Omega }(D^2u\cdot \nu )_{\partial \Omega }-\frac{\partial \Delta u}{\partial \nu }=\mu u, &{} \mathrm{on\ }\partial B, \end{array}\right. } \end{aligned}$$
(A.1)

are written in spherical coordinates as

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial ^2 u}{\partial r^2 }_{|_{r=1}}=\lambda \frac{\partial u}{\partial r }_{|_{r=1}} ,\\ -\frac{1}{r^2}{\Delta _S}\Big (\frac{\partial u}{\partial r}-\frac{u}{r}\Big )-\frac{\partial \Delta u}{\partial r}_{|_{r=1}}=\mu u_{|_{r=1}}, \end{array}\right. } \end{aligned}$$
(A.2)

where \(\Delta _S\) is the Laplace-Beltrami operator on the unit sphere. It is well known that the eigenfunctions can be written as a product of a radial part and an angular part (see e.g., [13] for details). In particular, the radial part is given in terms of power-type functions, while the angular part is written in terms of spherical harmonics. We have the following theorem.

Theorem A.1

Let \(\Omega =B\) be the unit ball in \(\mathbb R^N\) and \((\lambda ,\mu )\in \mathbb R^2\). Then there exists a non-trivial solution to problem \(\Delta ^2u=0\) on B with boundary conditions (A.1) if and only if there exists \(l\in \mathbb N_0\) such that \(\mathrm{det}M_l(\lambda ,\mu )=0\), where \(M_l(\lambda ,\mu )\) is the matrix defined by

$$\begin{aligned} M_l(\lambda ,\mu )= \begin{pmatrix} l(l-1-\lambda ) &{} (l+2)(l+1-\lambda ) \\ l(l+N-2)(l-1)-\mu &{} l(l(l-5)+N(l-1)-2)-\mu , \end{pmatrix} \end{aligned}$$
(A.3)

for all \(l\in \mathbb N_0\). If \((\lambda ,\mu )\in \mathbb R^2\) solves the equation \(\mathrm{det}M_l(\lambda ,\mu )=0\) for some \(l\in \mathbb N_0\), then the associated solutions can be written in the form

$$\begin{aligned} u_l(r,\theta )=\left( A_lr^{l}+B_lr^{l+2}\right) H_l(\theta ), \end{aligned}$$

where \((A_l,B_l)\in \mathbb R^2\) solves the linear system \(M_l(\lambda ,\mu )\cdot (A_l,B_l)=0\) and \(H_l(\theta )\) is a spherical harmonic of degree l in \(\mathbb R^N\).

Proof

It is well-known that the weak solutions of \(\Delta ^2u=0\) in the unit ball complemented with (A.1) are smooth (see e.g., [19, §2]). Moreover, we recall that any function u satisfying \(\Delta ^2u=0\) on the unit ball B along with the two homogeneous boundary conditions (A.2) can be written in spherical coordinates in the form

$$\begin{aligned} u_l(r,\theta )=(A r^l+B r^{l+2})H_l(\theta ), \end{aligned}$$
(A.4)

for \(l\in \mathbb N_0\), where \(A,B\in \mathbb R\) are arbitrary constants and \(H_l(\theta )\) is a spherical harmonic of degree l in \(\mathbb R^N\) (see e.g., [12, §5-6] for details). By using the explicit form (A.4) in (A.2), we find that the constants AB need to satisfy a homogeneous system of two linear equations, whose associated matrix is given by (A.3). Hence a non-trivial solution exists if and only if the determinant of (A.3) is zero. The rest of the statement is now a straightforward consequence. \(\square \)

By Theorem A.1 we immediately deduce the following characterization of the eigenvalues of (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)). Here by \(m_l\) we denote the dimension of the space of the spherical harmonics of degree \(l\in \mathbb N_0\) in \(\mathbb R^N\), that is

$$\begin{aligned} m_l=\frac{(2l+N-2)(l+N-3)!}{l!(N-2)!}. \end{aligned}$$

Corollary A.2

Any eigenvalue \(\lambda (\mu )\) of (\(\mathrm{BS}_{\mu }\)) on B satisfies the equation

$$\begin{aligned}&\lambda (\mu )\left( 2l^3+(N-1)l^2-(N-2)l-\mu \right) \nonumber \\&\quad =\left( 3l^4+2(N-2)l^3-(N+1)l^2-(N-2)l-(1+2l)\mu \right) , \end{aligned}$$
(A.5)

for some \(l\in \mathbb N_0\). Any eigenvalue \(\mu (\lambda )\) of (\(\mathrm{BS}_{\lambda }\)) on B satisfies the equation

$$\begin{aligned}&\mu (\lambda )\left( 2l+1-\lambda \right) \nonumber \\&\quad =\left( 3l^4+2(N-2)l^3-(N+1)l^2-(N-2)l-\left( 2l^3+(N-1)l^2-(N-2)l\right) \lambda \right) ,\nonumber \\ \end{aligned}$$
(A.6)

for some \(l\in \mathbb N_0\). The multiplicity of \(\mu (\lambda )\) and \(\lambda (\mu )\) corresponding to an index \(l\in \mathbb N_0\) equals the dimension \(m_l\) of the space of the spherical harmonics of degree l in \(\mathbb R^N\).

By using similar arguments, one can easily prove that the eigenvalues of problems (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)) on the unit ball B can also be determined explicitly.

Theorem A.3

Any eigenvalue \(\eta \) of problem (\(\mathrm{DBS}\)) on B is of the form

$$\begin{aligned} \eta =2l+1 \end{aligned}$$
(A.7)

for some \(l\in \mathbb N_0\) and its multiplicity equals the dimension \(m_l\) of the space of spherical harmonics of degree l in \(\mathbb R^N\).

Any eigenvalue \(\xi \) of problem (\(\mathrm{NBS}\)) on B is of the form

$$\begin{aligned} \xi =l(2l^2+(N-1)l-N+2), \end{aligned}$$
(A.8)

for some \(l\in \mathbb N_0\) and its multiplicity equals the dimension \(m_l\) of the space of spherical harmonics of degree l in \(\mathbb R^N\).

By combining Corollary A.2 and Theorem A.3 we can prove the following statement, where \(\lambda _{(l)}(\mu )\), \(\mu _{(l)}(\lambda )\), \(\eta _{(l)}\) and \(\xi _{(l)}\) denote the eigenvalues of (\(\mathrm{BS}_{\mu }\)), (\(\mathrm{BS}_{\lambda }\)), (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)), respectively, associated with spherical harmonics of order \(l\in \mathbb N_0\).

Theorem A.4

For all \(l\in \mathbb N\) and \(\mu \ne \xi _{(l)}\)

$$\begin{aligned} \lambda _{(l)}(\mu )=\frac{\left( 3l^4+2(N-2)l^3-(N+1)l^2-(N-2)l-\eta _{(l)}\mu \right) }{\left( \xi _{(l)}-\mu \right) }. \end{aligned}$$
(A.9)

For all \(l\in \mathbb N\) and \(\lambda \ne \eta _{(l)}\)

$$\begin{aligned} \mu _{(l)}(\lambda )=\frac{\left( 3l^4+2(N-2)l^3-(N+1)l^2-(N-2)l-\xi _{(l)}\lambda \right) }{\left( \eta _{(l)}-\lambda \right) }. \end{aligned}$$
(A.10)

Moreover, \(\lambda _{(0)}(\mu )=\eta _{(0)}=1\) for all \(\mu \in \mathbb R\) and \(\mu _{(0)}(\lambda )=\xi _{(0)}=0\) for all \(\lambda \in \mathbb R\).

Proof

According to the change of notation for eigenvalues, we denote by \(\lambda _{(l)}(\mu )\), \(\mu _{(l)}(\lambda )\), \(\eta _{(l)}\) and \(\xi _{(l)}\) the eigenvalues of problems (\(\mathrm{BS}_{\mu }\)), (\(\mathrm{BS}_{\lambda }\)), (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)) corresponding to the choice of \(l\in \mathbb N_0\) in (A.5), (A.6), (A.7) and (A.8). Each of such eigenvalues has multiplicity \(m_l\).

We note that (A.5) and (A.6) can be rewritten as

$$\begin{aligned} \lambda _{(l)}(\mu )\left( \xi _{(l)}-\mu \right) =\left( 3l^4+2(N-2)l^3-(N+1)l^2-(N-2)l-\eta _{(l)}\mu \right) \end{aligned}$$
(A.11)

and

$$\begin{aligned} \mu _{(l)}(\lambda )\left( \eta _{(l)}-\lambda \right) =\left( 3l^4+2(N-2)l^3-(N+1)l^2-(N-2)l-\xi _{(l)}\lambda \right) . \end{aligned}$$
(A.12)

If \(l\in \mathbb N\), then from (A.11) and (A.12) we deduce the validity of (A.9) when \(\mu \ne \xi _{(l)}\) and of (A.10) when \(\lambda \ne \eta _{(l)}\). If \(l=0\), the condition \(\mathrm{det}M_0(\lambda ,\mu )=0\) can be written in the form \(\mu (\lambda -1)=0\) which allows to conclude the proof. \(\square \)

We note that

$$\begin{aligned} \lim _{\mu \rightarrow -\infty }\lambda _{(l)}(\mu )=\eta _{(l)} \end{aligned}$$

and

$$\begin{aligned} \lim _{\lambda \rightarrow -\infty }\mu _{(l)}(\lambda )=\xi _{(l)} \end{aligned}$$

for all \(l\in \mathbb N_0\). This is coherent with Theorems 3.7 and 3.12. Moreover,

$$\begin{aligned} \lim _{\mu \rightarrow \xi _{(l)}^{\pm }}\lambda _{(l)}(\mu )=\pm \infty \end{aligned}$$

and

$$\begin{aligned} \lim _{\lambda \rightarrow \eta _{(l)}^{\pm }}\mu _{(l)}(\lambda )=\pm \infty \end{aligned}$$

for all \(l\in \mathbb N\). We have shown that the branches of eigenvalues \(\lambda _{(l)}(\mu )\) and \(\mu _{(l)}(\lambda )\) are analytic functions of their parameters on \(\mathbb R\setminus \{\xi _{(l)}\}_{l\in \mathbb N_0}\) and \(\mathbb R\setminus \{\eta _{(l)}\}_{l\in \mathbb N_0}\) respectively. In particular, the branch \(\lambda _{(l)}(\mu )\) is a equilateral hyperbole with \(\eta _{(l)}\) as horizontal asymptote and \(\xi _{(l)}\) as vertical asymptote, if \(l\ge 1\), while it is coincides with \(\left\{ (\mu ,1):\mu \in \mathbb R\right\} \) for \(l=0\). The branch \(\mu _{(l)}(\lambda )\) is a equilateral hyperbole with \(\xi _{(l)}\) as horizontal asymptote and \(\eta _{(l)}\) as vertical asymptote, if \(l\ge 1\), while it is coincides with \(\left\{ (\lambda ,0):\mu \in \mathbb R\right\} \) for \(l=0\). The situation is illustrated in Fig. 1.

Fig. 1
figure1

Eigenvalues \(\mu _{(l)}(\lambda )\) of (\(\mathrm{BS}_{\lambda }\)) as functions of \(\lambda \) (the parameter \(\lambda \) correspond to the abscissa). Vertical asymptotes are the eigenvalues \(\eta _{(l)}\) of (\(\mathrm{DBS}\)). Horizontal asymptotes are the eigenvalues \(\xi _{(l)}\) of (\(\mathrm{NBS}\)). A reflection along the angle bisector of the first and third quadrant gives the eigenvalues \(\lambda _{(l)}(\mu )\) of (\(\mathrm{BS}_{\mu }\)) as functions of \(\mu \)

Asymptotic formulas

It is proved in [28, 29] that if \(\Omega \) is a bounded domain in \(\mathbb R^N\) with \(C^{\infty }\) boundary, then the eigenvalues of problems (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)) with \(\sigma =1\) satisfy the following asymptotic laws

$$\begin{aligned} \eta _j\sim \frac{4\pi }{\omega _{N-1}^{\frac{1}{N-1}}}\left( \frac{j}{|\partial \Omega |}\right) ^{\frac{1}{N-1}}, \end{aligned}$$
(B.1)

and

$$\begin{aligned} \xi _j\sim \frac{16\pi ^3}{\omega _{N-1}^{\frac{3}{N-1}}}\left( \frac{j}{|\partial \Omega |}\right) ^{\frac{3}{N-1}}, \end{aligned}$$
(B.2)

as \(j\rightarrow \infty \). Here \(\omega _{N-1}\) denotes the volume of the unit ball in \(\mathbb R^{N-1}\).

We state the following theorem, whose proof is omitted since it follows exactly the same lines as those of Theorems 1.1 and 1.2 of [29].

Theorem B.1

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{\infty }\). Then formulas (B.1) and (B.2) hold for all \(\sigma \in \big (-\frac{1}{N-1},1\big )\). Moreover, the two following asymptotic formulas hold for the eigenvalues of problems (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\))

$$\begin{aligned} \lambda _j(\mu )\sim \frac{3\pi }{\omega _{N-1}^{\frac{1}{N-1}}}\left( \frac{j}{|\partial \Omega |}\right) ^{\frac{1}{N-1}}, \end{aligned}$$
(B.3)

and

$$\begin{aligned} \mu _j(\lambda )\sim \frac{12\pi ^3}{\omega _{N-1}^{\frac{3}{N-1}}}\left( \frac{j}{|\partial \Omega |}\right) ^{\frac{3}{N-1}} \end{aligned}$$
(B.4)

as \(j\rightarrow \infty \).

We note that the principal term in the asymptotic expansions of the eigenvalues depends neither on the Poisson’s ratio \(\sigma \) nor on \(\mu \) or \(\lambda \). However, lower order terms have to depend on \(\mu \) and \(\lambda \), since, as \(\mu ,\lambda \rightarrow -\infty \), \(\lambda _j(\mu )\rightarrow \eta _j\) and \(\mu _j(\lambda )\rightarrow \xi _j\), and asymptotic formulas of \(\lambda _j(\mu )\) and \(\eta _j\), and of \(\mu _j(\lambda )\) and \(\xi _j\), differ from a factor \(\frac{3}{4}\).

Remark B.2

We remark that the approach used in [29] requires that the boundary of \(\Omega \) is of class \(C^{\infty }\). However, as proved by another technique in [28], the asymptotic formulas for the eigenvalues of (\(\mathrm{DBS}\)) and (\(\mathrm{NBS}\)) when \(\sigma =1\) hold when \(\Omega \) is of class \(C^{2}\).

We now show that in the case of the unit ball B in \(\mathbb R^N\) it is possible to recover formulas (B.1), (B.2), (B.3) and (B.4) by using the explicit computations in Appendix A.

Note that for a fixed \(l\in \mathbb N_0\), the dimension of the space of spherical harmonics of degree less or equal than l is \(\frac{(2l+N-1)(N+l-2)!}{l!(N-1)!}\). By (A.8) we deduce that

$$\begin{aligned} \xi _j=l(2l^2+(N-1)l-N+2) \end{aligned}$$
(B.5)

whenever \(j\in \mathbb N\) is such that

$$\begin{aligned} \frac{(2l+N-3)(N+l-3)!}{(l-1)!(N-1)!}<j\le \frac{(2l+N-1)(N+l-2)!}{l!(N-1)!} \end{aligned}$$
(B.6)

Moreover,

$$\begin{aligned}&\lim _{l\rightarrow +\infty }\frac{(2l+N-3)(N+l-3)!}{(l-1)!(N-1)!}\div \frac{2l^{N-1}}{(N-1)!}\\&\quad =\lim _{l\rightarrow +\infty }\frac{(2l+N-1)(N+l-2)!}{l!(N-1)!}\div \frac{2l^{N-1}}{(N-1)!}=1. \end{aligned}$$

From (B.5) and (B.6) we deduce that

$$\begin{aligned} \xi _{j}\sim 2^{\frac{N-4}{N-1}}(N-1)!^{\frac{3}{N-1}}j^{\frac{3}{N-1}}\,,\ \ \ \mathrm{as\ }j\rightarrow +\infty . \end{aligned}$$

We note that this is exactly (B.2). Indeed, recalling that \(|\partial B|=N\omega _N\), a standard computation shows that

$$\begin{aligned} 2^{\frac{N-4}{N-1}}(N-1)!^{\frac{3}{N-1}}=\frac{16\pi ^3}{\omega _{N-1}^{\frac{3}{N-1}}}\cdot \frac{1}{|\partial \Omega |^{\frac{3}{N-1}}}, \end{aligned}$$

for which it is useful to note that

$$\begin{aligned} \omega _N\omega _{N-1}=\frac{2^N\pi ^{N-1}}{N!}. \end{aligned}$$

In the same way we verify that

$$\begin{aligned}&\eta _j\sim 2^{\frac{N-2}{N-1}}(N-1)!^{\frac{1}{N-1}}j^{\frac{1}{N-1}},\\&\quad \lambda _j(\mu )\sim \frac{3}{4}2^{\frac{N-2}{N-1}}(N-1)!^{\frac{1}{N-1}}j^{\frac{1}{N-1}}, \end{aligned}$$

and

$$\begin{aligned} \mu _j(\lambda )\sim \frac{3}{4}2^{\frac{N-4}{N-1}}(N-1)!^{\frac{3}{N-1}}j^{\frac{3}{N-1}}, \end{aligned}$$

as \(j\rightarrow +\infty \), and these asymptotic formulas correspond to formulas (A.7), (A.5) and (A.6).

The (\(\mathrm{BS}_{\mu }\)) and (\(\mathrm{BS}_{\lambda }\)) problems for \(\mu >0\) and \(\lambda >\eta _1\)

In this section we briefly discuss problem (\(\mathrm{BS}_{\mu }\)) for \(\mu >0\) and problem (\(\mathrm{BS}_{\lambda }\)) for \(\lambda >\eta _1\).

We begin with problem (\(\mathrm{BS}_{\mu }\)). Assume that \(\mu \in \mathbb R\) is such that \(\xi _j<\mu <\xi _{j+1}\) for some \(j\in \mathbb N\). Recall that \(\xi _j\) denote the eigenvalues of problem (\(\mathrm{NBS}\)). We denote by \(U_j\) the subspace of \(H^2(\Omega )\) generated by all eigenfunctions \(u_i\) associated with the eigenvalues \(\xi _i\) with \(i\le j\) and we set

$$\begin{aligned} U_j^{\perp }=\left\{ u\in H^2(\Omega ):\mathcal Q_{\mu ,D}(u,\varphi )=0\,,\ \ \ \forall \varphi \in U_j\right\} . \end{aligned}$$

The space \(U_j^{\perp }\) is a closed subspace of \(H^2(\Omega )\). We have the following result.

Theorem C.1

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\) and assume that \(\xi _j<\mu <\xi _{j+1}\) for some \(j\in \mathbb N\). Then

$$\begin{aligned} H^2(\Omega )=U_j\oplus U_j^{\perp }. \end{aligned}$$
(C.1)

Moreover, there exists \(b\ge 0\) such that the quadratic form \(\mathcal Q_{\mu ,D}(u,v)+b(\gamma _1(u),\gamma _1(v))_{\partial \Omega }\), \(u,v\in H^2(\Omega )\), is coercive on \(U_j^{\perp }\).

Note that the decomposition (C.1) is not straightforward since \(\mathcal Q_{\mu ,D}\) does not define a scalar product for \(\mu >0\). However, in order to prove Theorem C.1 one can easily adapt the analogous proof in [27, Theorem 5.1] but we omit the details.

We observe that, whenever \(u\in U_j^{\perp }\), one can consider only test functions \(\varphi \in U_j^{\perp }\) in the weak formulation of (3.4). Indeed, adding to \(\varphi \) a test function \(\tilde{\varphi }\in U_j\) leaves both sides of the equation unchanged. Hence one can perform the same analysis as in the case \(\mu <0\) and state an analogous version of Theorem 3.3 with the space \(H^2(\Omega )\) replaced by \(U_j^{\perp }\). We leave this to the reader.

Remark C.2

If \(\mu =\xi _j\) for some \(j\in \mathbb N\) the situation is more involved and is not analyzed here. However, if we assume that \(\xi _j\) is an eigenvalue of (3.3) of multiplicity m such that

$$\begin{aligned} \mathcal Q_{\sigma }(u,\varphi )=\xi _j(\gamma _0(u),\gamma _0(\varphi ))_{\partial \Omega }\,,\ \ \ \forall \varphi \in H^2(\Omega ), u\in U_{\xi _j}, \end{aligned}$$
(C.2)

where \(U_{\xi _j}\) is the eigenspace generated by all the eigenfunctions \(\{u_j^1,...,u_j^m\}\) in \(\mathcal H^2_{0,N}(\Omega )\) associated with \(\xi _j\), the problem becomes simpler. Indeed, any function \(w\in H^2(\Omega )\) can be written in the form \(w=u+v\), where \(u\in U_{\xi _j}\) and \(v\in H^2_{\xi _j}(\Omega )\), where

$$\begin{aligned} H^2_{\xi _j}(\Omega ):=\left\{ v\in H^2(\Omega ): (\gamma _0(v),\gamma _0(u_j^i))_{\partial \Omega }=0\mathrm{\ for\ all\ }i=1,...,m\right\} . \end{aligned}$$

Hence, whenever \(u\in H^2_{\xi _j}(\Omega )\), one can consider only test functions in \(\varphi \in H^2_{\xi _j}(\Omega )\) in the weak formulation (3.4) with \(\mu =\xi _j\). In fact, adding to \(\varphi \) a function \(\tilde{\varphi }\in U_{\xi _j}\) leaves both sides of the equation unchanged. Thus we can perform the same analysis of Theorem C.1 with \(H^2(\Omega )\) replaced by \(H^2_{\xi _j}(\Omega )\).

Note that equation (C.2) is satisfied with \(\xi _1=0\) and u a constant function.

We also have an analogous result for problem (\(\mathrm{BS}_{\lambda }\)) with \(\lambda >\eta _1\). Assume that \(\lambda \in \mathbb R\) is such that \(\eta _j<\lambda <\eta _{j+1}\) for some \(j\in \mathbb N\). Recall that \(\eta _j\) denote the eigenvalues of problem (\(\mathrm{DBS}\)). We denote by \(V_j\) the subspace of \(H^2(\Omega )\) generated by all eigenfunctions \(v_i\) associated with the eigenvalues \(\eta _i\) with \(i\le j\) and we set

$$\begin{aligned} V_j^{\perp }=\left\{ v\in H^2(\Omega ):\mathcal Q_{\lambda ,N}(v,\varphi )=0\,,\ \ \ \forall \varphi \in V_j\right\} . \end{aligned}$$

The space \(V_j^{\perp }\) is a closed subspace of \(H^2(\Omega )\). We have the following result.

Theorem C.3

Let \(\Omega \) be a bounded domain in \(\mathbb R^N\) of class \(C^{0,1}\) and assume that \(\eta _j<\lambda <\eta _{j+1}\). Then

$$\begin{aligned} H^2(\Omega )=V_j\oplus V_j^{\perp }. \end{aligned}$$
(C.3)

Moreover, there exists \(b\ge 0\) such that the quadratic form \(\mathcal Q_{\lambda ,N}(u,v)+b(\gamma _0(u),\gamma _0(v))_{\partial \Omega }\), \(u,v\in H^2(\Omega )\), is coercive on \(V_j^{\perp }\).

We observe that, whenever \(v\in V_j^{\perp }\), one can consider only test functions \(\varphi \in V_j^{\perp }\) in the weak formulation of (3.14). In fact, adding to \(\varphi \) a test function \(\tilde{\varphi }\in V_j\) leaves both sides of the equation unchanged. Hence one can perform the same analysis as in the case \(\lambda <\eta _1\) and state an analogous version of Theorem 3.10 with the space \(H^2(\Omega )\) replaced by \(V_j^{\perp }\). We leave this to the reader.

Remark C.4

As in Remark C.2, one can treat the case \(\lambda =\eta _j\) for some j in the special situation when \(\eta _j\) is an eigenvalue of (3.1) of multiplicity m such that

$$\begin{aligned} \mathcal Q_{\sigma }(v,\varphi )=\eta _j(\gamma _1(v),\gamma _1(\varphi ))\,,\ \ \ \forall \varphi \in H^2(\Omega ), v\in V_{\eta _j}, \end{aligned}$$

where \(V_{\eta _j}\) is the eigenspace generated by all the eigenfunctions \(\{v_j^1,...,v_j^m\}\) in \(\mathcal H^2_{0,D}(\Omega )\) associated with \(\eta _j\). This happens in the case of the unit ball with \(\eta _1=1\). We observe that any function \(w\in H^2(\Omega )\) can be written in the form \(w=v+u\), where \(v\in V_{\eta _j}\) and \(u\in H^2_{\eta _j}(\Omega )\), where

$$\begin{aligned} H^2_{\eta _j}(\Omega ):=\left\{ u\in H^2(\Omega ): (\gamma _1(u),\gamma _1(v_j^i))_{\partial \Omega }=0\mathrm{\ for\ all\ }i=1,...,m\right\} . \end{aligned}$$

Hence, whenever \(v\in H^2_{\eta _j}(\Omega )\), one can consider in the weak formulation (3.14) with \(\lambda =\eta _j\) only test functions in \(\varphi \in H^2_{\eta _j}(\Omega )\). In fact, adding to \(\varphi \) a function \(\tilde{\varphi }\in V_{\eta _j}\) leaves both sides of the equation unchanged. Thus we can perform the same analysis of Theorem C.3 with \(H^2(\Omega )\) replaced by \(H^2_{\eta _j}(\Omega )\).

We conclude this section with a few more remarks. We have observed that the eigenvalue \(\mu =0\) is always an eigenvalue of (3.14) when \(\lambda <\eta _1\), and that the constant functions belong to the eigenspace associated with \(\mu =0\), and in particular belong to the space \(\mathcal H^2_{0,N}(\Omega )\) and are eigenfunctions associated with the first eigenvalue \(\xi _1=0\) of problem (\(\mathrm{NBS}\)). Such a situation may occur also for other eigenvalues, as well as for problem (3.4) (as we have seen in the case of the ball in \(\mathbb R^N\) with the eigenvalue \(\lambda (\mu )=\eta _1=1\) for all \(\mu \in \mathbb R\)). The following lemma clarifies this phenomenon.

Lemma C.5

Let \(j\in \mathbb N\). Then one of the following two alternatives occurs for problem (3.4):

  1. i)

    \(\lambda _j(\mu )<\eta _j\) for all \(\mu \in (-\infty ,0)\);

  2. ii)

    there exists \(\mu _0\in (-\infty ,0)\) such that \(\lambda _j(\mu _0)=\eta _j\). In this case, \(\eta _j=\lambda _j(\mu )\) for all \(\mu \in (-\infty ,\mu _0]\) and \(\eta _j\) is an eigenvalue of problem (3.4) for any \(\mu \in \mathbb R\).

Proof

From the Min-Max Principles (3.2) and (3.8) we have that \(\lambda _j(\mu )\le \eta _j\) for all \(\mu \in (-\infty ,0)\), \(j\in \mathbb N\). Assume now that there exists \(\mu _0\in (-\infty ,0)\) such that \(\lambda _j(\mu _0)=\eta _j\). Again from (3.2) and (3.8) we deduce that we can choose an eigenfunction \(v_{j,\mu _0}\) associated with \(\lambda _j(\mu _0)\) which belong to \(\mathcal {H}^2_{0,D}(\Omega )\) and which coincide with an eigenfunction \(u_j\) of problem (3.1) associated with \(\eta _j\). In fact the eigenfunctions associated with \(\lambda _j(\mu )\) are exactly the functions realizing the equality in (3.8). We also note that

$$\begin{aligned} \mathcal Q_{\sigma }(u_j,\varphi )=\mathcal Q_{\mu ,D}(u_j,\varphi )\,,\ \ \ \forall \varphi \in H^2(\Omega ),\mu \in \mathbb R, \end{aligned}$$

hence

$$\begin{aligned} \mathcal Q_{\mu ,D}(u_j,\varphi )=\eta _j(u_j,\varphi )_{\partial \Omega }\,,\ \ \ \forall \varphi \in H^2(\Omega ),\mu \in \mathbb R, \end{aligned}$$

hence \(\eta _j\) is an eigenvalue of (3.4) for all \(\mu \in \mathbb R\) and in particular \(\eta _j=\lambda _j(\mu )\) for all \(\mu \in (-\infty ,\mu _0]\). This concludes the proof. \(\square \)

In the same way one can prove the following.

Lemma C.6

Let \(j\in \mathbb N\). Then one of the following two alternatives occurs for problem (3.14):

  1. i)

    \(\mu _j(\lambda )<\xi _j\) for all \(\lambda \in (-\infty ,\eta _1)\);

  2. ii)

    there exists \(\lambda _0\in (-\infty ,\eta _1)\) such that \(\mu _j(\lambda _0)=\xi _j\). In this case, \(\xi _j=\mu _j(\lambda )\) for all \(\lambda \in (-\infty ,\lambda _0]\) and \(\xi _j\) is an eigenvalue of problem (3.14), for any \(\lambda \in \mathbb R\).

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Lamberti, P.D., Provenzano, L. On the explicit representation of the trace space \(H^{\frac{3}{2}}\) and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems. Rev Mat Complut (2021). https://doi.org/10.1007/s13163-021-00385-z

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Keywords

  • Bi-Laplacian
  • Steklov boundary conditions
  • Multi-parameter eigenvalue problems
  • Biharmonic Steklov eigenvalues
  • Fourier series
  • Trace spaces

Mathematics Subject Classification

  • 35J40
  • 35P10
  • 46E35