# On time-periodic solutions to parabolic boundary value problems

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## Abstract

Time-periodic solutions to partial differential equations of parabolic type corresponding to an operator that is elliptic in the sense of Agmon–Douglis–Nirenberg are investigated. In the whole- and half-space case we construct an explicit formula for the solution and establish coercive \(L^{p}\) estimates. The estimates generalize a famous result of Agmon, Douglis and Nirenberg for elliptic problems to the time-periodic case.

## Mathematics Subject Classification

Primary 35B10 35B45 35K25## 1 Introduction

*m*and \(B_1,\ldots ,B_m\) satisfy an appropriate complementing boundary condition. The domain \(\Omega \) is either the whole-space, the half-space or a bounded domain, and \(\mathbb {R}\) denotes the time-axis. The solutions \(u(t,x)\) correspond to time-periodic data \(f(t,x)\) and \(g_j(t,x)\) of the same (fixed) period \(T>0\). Using the simple projections

*purely oscillatory*problem

The decomposition (1.2)–(1.3) is essential as the two problems have substantially different properties. In particular, we shall show in the whole- and half-space case that the principle part of the linear operator in the purely oscillatory problem (1.3) is a homeomorphism in a canonical setting of time-periodic Lebesgue–Sobolev spaces. This is especially remarkable since the elliptic problem (1.2) does not satisfy this property. Another remarkable characteristic of (1.3) is that the \(L^{p}\) theory we shall develop for this problem leads directly to a similar \(L^{p}\) theory, sometimes referred to as maximal regularity, for the parabolic initial-value problem associated to (1.1).

*m*and \(m_j\) (\(j=1,\ldots ,m\)), respectively, with no restrictions other than \(m\in \mathbb {N}\) and \(m_j\in \mathbb {N}_0\). We denote the principle part of the operators by

### Definition 1.1

(*Properly Elliptic*) The operator \(A^H\) is said to be *properly elliptic* if for all \(x\in \Omega \) and all \(\xi \in \mathbb {R}^n\setminus \{0\}\) it holds \(A^H(x,\xi )\ne 0\), and for all \(x\in \Omega \) and all linearly independent vectors \({\zeta },\xi \in \mathbb {R}^n\) the polynomial \(P(\tau ):= A^H(x,{\zeta }+\tau \xi )\) has *m* roots in \(\mathbb {C}\) with positive imaginary part, and *m* roots in \(\mathbb {C}\) with negative imaginary part.

Ellipticity, however, does not suffice to establish maximal \(L^{p}\) regularity for the time-periodic problem. We thus recall Agmon’s condition, also known as parameter ellipticity.

### Definition 1.2

(*Agmon’s Condition*) Let \(\theta \in [-\pi ,\pi ]\). A properly elliptic operator \(A^H\) is said to satisfy Agmon’s condition on the ray \({{\mathrm{e}}}^{i\theta }\) if for all \(x\in \Omega \) and all \(\xi \in \mathbb {R}^n\setminus \{0\}\) it holds \(A^H(x,\xi )\notin \{r{{\mathrm{e}}}^{i\theta }\ |\ r\ge 0\}\).

If \(A^H\) satisfies Agmon’s condition on the ray \({{\mathrm{e}}}^{i\theta }\), then, since the roots of a polynomial depend continuously on its coefficients, the polynomial \(Q(\tau ):=-r{{\mathrm{e}}}^{i\theta }+A^H(x,{\zeta }+\tau \xi )\) has *m* roots \(\tau _h^+(r{{\mathrm{e}}}^{i\theta },x,{\zeta },\xi )\in \mathbb {C}\) with positive imaginary part, and *m* roots \(\tau _h^-(r{{\mathrm{e}}}^{i\theta },x,{\zeta },\xi )\in \mathbb {C}\) with negative imaginary part (\(h=1,\ldots ,m\)). Consequently, the following assumption on the operator \((A^H,B^H_1,\ldots ,B^H_m)\) is meaningful.

### Definition 1.3

*Agmon’s Complementing Condition*) Let \(\theta \in [-\pi ,\pi ]\). If \(A^H\) is a properly elliptic operator, then \((A^H,B^H_1,\ldots ,B^H_m)\) is said to satisfy Agmon’s complementing condition on the ray \({{\mathrm{e}}}^{i\theta }\) if:

- (i)
\(A^H\) satisfies Agmon’s condition on the ray \({{\mathrm{e}}}^{i\theta }\).

- (i)
For all \(x\in \partial \Omega \), all pairs \({\zeta },\xi \in \mathbb {R}^n\) with \({\zeta }\) tangent to \(\partial \Omega \) and \(\xi \) normal to \(\partial \Omega \) at

*x*, and all \(r\ge 0\), let \(\tau _h^+(r{{\mathrm{e}}}^{i\theta },x,{\zeta },\xi )\in \mathbb {C}\) (\(h=1,\ldots ,m\)) denote the*m*roots of the polynomial \({Q(\tau ):=-r{{\mathrm{e}}}^{i\theta }+A^H(x,{\zeta }+\tau \xi )}\) with positive imaginary part. The polynomials \(P_j(\tau ):=B^H_j(x,{\zeta }+\tau \xi )\) (\(j=1,\ldots ,m\)) are linearly independent modulo the polynomial \(\Pi _{h=1}^m \big (\tau -\tau _h^+(r{{\mathrm{e}}}^{i\theta },x,{\zeta },\xi )\big )\).

*parameter ellipticity*. We note that it is equivalent to the

*Lopatinskiĭ-Shapiro condition*, see Remark 1.7 below. The condition was introduced in order to identify the additional requirements on the differential operators needed to extend the result of Agmon, Douglis and Nirenberg [3] from the elliptic case to the corresponding parabolic initial-value problem. The theorem of Agmon, Douglis and Nirenberg [3] requires \((A^H,B^H_1,\ldots ,B^H_m)\) to satisfy Agmon’s complementing condition only at the origin (

*not*on a full ray), in which case \((A^H,B^H_1,\ldots ,B^H_m)\) is said to be

*elliptic in the sense of Agmon–Douglis–Nirenberg*. Analysis of the associated initial-value problem relies heavily on properties of the resolvent equation

*without*the use of either bounded imaginary powers or the notion of \({\mathcal R}\)-boundedness; see Remark 1.6 below. Under the assumption that \((A^H,B^H_1,\ldots ,B^H_m)\) generates an analytic semi-group, maximal regularity for the parabolic initial-value problem follows almost immediately as a corollary from our main theorem. We emphasize that our main theorem of maximal regularity for the time-periodic problem does

*not*require the principle part of \((A,B_1,\ldots ,B_m)\) to generate an analytic semi-group. As a novelty of the present paper, and in contrast to the initial-value problem, we establish that maximal \(L^{p}\) regularity for the time-periodic problem requires Agmon’s complementing condition to be satisfied only on the two rays with \(\theta =\pm \,\frac{\pi }{2}\), that is, only on the imaginary axis.

The references above to the theory of maximal \(L^{p}\) regularity for parabolic initial-value problems would not be complete without mention of the extensive work of Solonnikov on initial-value problems for parabolic systems; see [23] and the references therein. The investigation of systems requires a more involved definition of parabolicity and complementary condition than Definition 1.1–1.3, but the arguments towards an \(L^{p}\) theory follow similar ideas as in the scalar case. As pointed out by Wang [28], the approach of Solonnikov can be reduced to an argument based on Fourier multipliers. This rationale was also proposed by Arkeryd [7] in his study of elliptic boundary value problems and will also be used in our approach in the time-periodic case.

Our main theorem for the purely oscillatory problem (1.3) concerns the half-space case and the question of existence of a unique solution satisfying a coercive \(L^{p}\) estimate in the Sobolev space \(W^{1,2m,p}_{\mathrm {per}}(\mathbb {R}\times \mathbb {R}^n_+)\) of time-periodic functions on the time-space domain \(\mathbb {R}\times \mathbb {R}^n_+\). We refer to Sect. 2 for definitions of the function spaces.

### Theorem 1.4

Our proof of Theorem 1.4 contains two results that are interesting in their own right. Firstly, we establish a similar assertion in the whole-space case. Secondly, we provide an explicit formula for the solution; see (3.8) below. Moreover, our proof is carried out fully in a setting of time-periodic functions and follows an argument adopted from the elliptic case. This is remarkable in view of the fact that analysis of time-periodic problems in existing literature typically is based on theory for the corresponding initial-value problem; see for example [18]. A novelty of our approach is the introduction of suitable tools from abstract harmonic analysis that allow us to give a constructive proof and avoid completely the classical indirect characterizations of time-periodic solutions as fixed points of a Poincaré map, that is, as special solutions to the corresponding initial-value problem. The circumvention of the initial-value problem also enables us to avoid having to assume Agmon’s condition for all \({|\theta |}\ge \frac{\pi }{2}\) and instead carry out our investigation under the weaker condition that Agmon’s condition is satisfied only for \(\theta =\pm \frac{\pi }{2}\).

We shall briefly describe the main ideas behind the proof of Theorem 1.4. We first consider the problem in the whole space \(\mathbb {R}\times \mathbb {R}^n\) and replace the time axis \(\mathbb {R}\) with the torus \({\mathbb T}:=\mathbb {R}/T\mathbb {Z}\) in order to reformulate the \(T\)-time-periodic problem as a partial differential equation on the locally compact abelian group \(G:={\mathbb T}\times \mathbb {R}^n\). Utilizing the Fourier transform \(\mathscr {F}_G\) associated to \(G\), we obtain an explicit representation formula for the time-periodic solution. Since \(\mathscr {F}_G=\mathscr {F}_{\mathbb T}\circ \mathscr {F}_{\mathbb {R}^n}\), this formula simply corresponds to a Fourier series expansion in time of the solution and subsequent Fourier transform in space of all its Fourier coefficients. While it is relatively easy to obtain \(L^{p}\) estimates (in space) for each Fourier coefficient separately, it is highly non-trivial to deduce from these individual estimates an \(L^{p}\) estimate in space *and* time via the corresponding Fourier series. Instead, we turn to the representation formula given in terms of \(\mathscr {F}_G\) and show that the corresponding Fourier multiplier defined on the dual group \(\widehat{G}\) is an \(L^{p}(G)\) multiplier. For this purpose, we use the so-called *Transference Principle* for Fourier multipliers in a group setting, and obtain the necessary estimate in the whole-space case. In the half-space case, Peetre [22] and Arkeryd [7] utilized the Paley–Wiener Theorem in order to construct a representation formula for solutions to elliptic problems; see also [26, Section 5.3]. We adapt their ideas to our setting and establish \(L^{p}\) estimates from the ones already obtained in the whole-space case.

### Theorem 1.5

Since time-independent functions are trivially also time-periodic, we have \(W^{2m,p}(\Omega )\subset W^{1,2m,p}_{\mathrm {per}}(\mathbb {R}\times \Omega )\). If estimate (1.9) is restricted to functions in \(W^{2m,p}(\Omega )\), Theorem 1.5 reduces to the classical theorem of Agmon–Douglis–Nirenberg [3], which has played a fundamental role in the analysis of elliptic boundary value problems for more then half a century now. This classical theorem for scalar equations was extended to systems in [4]. We shall only treat scalar equations in the following, but will address systems in future works.

We briefly return to the decomposition (1.2)–(1.3). It is well-known in the bounded domain case that ellipticity of (1.2) in the sense of Agmon–Douglis–Nirenberg is equivalent to the corresponding linear operator being Fredholm in the setting of classical Sobolev spaces. From Theorem 1.4 and the similar assertion in the whole-space case, which as mentioned above shall also be provided, one can show that also the operator of the purely oscillatory problem (1.3) is Fredholm in the setting of time-periodic Sobolev spaces. Indeed, since we show that the operator is a homeomorphism in the whole- and half-space cases, a localization argument (see for example [29, Proof of Theorem 13.1] or [30, Proof of Theorem 9.32]) yields existence of a left and right regularizer in the bounded domain case, which in turn implies the Fredholm property. Since both the elliptic and purely oscillatory problem possess the Fredholm property, so does the full time-periodic problem on bounded domains. Due to the work of Geymonat [15], a comprehensive Fredholm theory is available for the elliptic problem (1.2). Since our proof of Theorem 1.5 successfully demonstrates that time-periodic problems can be approached in much the same way as elliptic problems, it seems likely that a similar comprehensive Fredholm theory can be developed for the purely oscillatory problem (1.3). Although we shall leave this investigation to future works, we note that the Fredholm properties of the operator in (1.3) will in general be different from the Fredholm properties of the elliptic problem (1.2). In fact, the simple example of the Laplace equation with a Neumann boundary condition in a bounded domain shows that the defect numbers of the two problems can be different. This observation further underlines the importance of the decomposition (1.2)–(1.3).

Time-periodic problems of parabolic type have been investigated in numerous articles over the years, and it would be too far-reaching to list them all here. We mention only the article of Liebermann [18], the recent article by Geissert, Hieber and Nguyen [14], as well as the monographs [16, 27], and refer the reader to the references therein. Finally, we mention the article [17] by the present authors in which some of the ideas utilized in the following were introduced in a much simpler setting.

### Remark 1.6

### Remark 1.7

*x*, all \(r\ge 0\) and all \(g=(g_0,\ldots ,g_{m-1})\in \mathbb {C}^m\) the system of ordinary differential equations

## 2 Preliminaries and notation

### 2.1 Notation

Unless otherwise indicated, *x* denotes an element in \(\mathbb {R}^n\) and \(x':=(x_1,\ldots ,x_{n-1})\in \mathbb {R}^{n-1}\). The same notation is employed for \(\xi \in \mathbb {R}^n\) and \(\xi ':=(\xi _1,\ldots ,\xi _{n-1})\in \mathbb {R}^{n-1}\).

We denote by \(\mathbb {C}_+:=\{z\in \mathbb {C}\ |\ {{\mathrm{Im}}}(z)>0\}\) and \(\mathbb {C}_-:=\{z\in \mathbb {C}\ |\ {{\mathrm{Im}}}(z)<0\}\) the upper and lower complex plane, respectively.

The notation \(\partial _j:=\partial _{x_j}\) is employed for partial derivatives with respect to spatial variables. Throughout, \(\partial _t\) shall denote the partial derivative with respect to the time variable.

*g*

*parabolically*\(\alpha \)

*-homogeneous*if \(\lambda ^\alpha g(\eta ,\xi ) = g(\lambda ^{2m}\eta ,\lambda \xi )\) for all \(\lambda >0\).

### 2.2 Paley–Wiener theorem

### Definition 2.1

### Proposition 2.2

(Paley–Wiener Theorem) Let \(f\in L^{2}(\mathbb {R})\). Then \({{\mathrm{supp}}}f\subset \overline{\mathbb {R}_+}\) if and only if \(\widehat{f}\in \mathscr {H}_+^2\). Moreover, \({{\mathrm{supp}}}f\subset \overline{\mathbb {R}_-}\) if and only if \(\widehat{f}\in \mathscr {H}_-^2\).

### Proof

See for example [31, Theorems VI.4.1 and VI.4.2]. \(\square \)

### 2.3 Time-periodic function spaces

### 2.4 Function spaces and the torus group setting

We shall further introduce a setting of function spaces in which the time axis \(\mathbb {R}\) in the underlying domains is replaced with the torus \({\mathbb T}:=\mathbb {R}/T\mathbb {Z}\). In such a setting, all functions are inherently \(T\)-time-periodic. We shall therefore never have to verify periodicity of functions a posteriori, and it will always be clear in which sense the functions are periodic.

*purely oscillatory*.

### 2.5 Schwartz–Bruhat spaces and distributions

When the spatial domain is the whole-space \(\mathbb {R}^n\), we employ the notation \({G:={\mathbb T}\times \mathbb {R}^n}\). Equipped with the quotient topology via \(\pi \), \(G\) becomes a locally compact abelian group. Clearly, the \(L^{p}(G)\) space corresponding to the Haar measure on \(G\), appropriately normalized, coincides with the \(L^{p}({\mathbb T}\times \mathbb {R}^n)\) space introduced in the previous section.

We identify \(G\)’s dual group by \(\widehat{G}=\frac{2\pi }{T}\mathbb {Z}\times \mathbb {R}^n\) by associating \((k,\xi )\in \frac{2\pi }{T}\mathbb {Z}\times \mathbb {R}^n\) with the character \(\chi :G\rightarrow \mathbb {C},\ \chi (x,t):={{\mathrm{e}}}^{ix\cdot \xi +ik t}\). By default, \(\widehat{G}\) is equipped with the compact-open topology, which in this case coincides with the product of the discrete topology on \(\frac{2\pi }{T}\mathbb {Z}\) and the Euclidean topology on \(\mathbb {R}^n\). The Haar measure on \(\widehat{G}\) is simply the product of the Lebesgue measure on \(\mathbb {R}^n\) and the counting measure on \(\frac{2\pi }{T}\mathbb {Z}\).

The topological dual space \(\mathscr {S^\prime }(G)\) of \(\mathscr {S}(G)\) is referred to as the space of tempered distributions on \(G\). Observe that both \(\mathscr {S}(G)\) and \(\mathscr {S^\prime }(G)\) remain closed under multiplication by smooth functions that have at most polynomial growth with respect to the spatial variables. For a tempered distribution \(u\in \mathscr {S^\prime }(G)\), distributional derivatives \(\partial _t^\alpha \partial _x^\beta u\in \mathscr {S^\prime }(G)\) are defined by duality in the usual manner. Also the support \({{\mathrm{supp}}}u\) is defined in the classical way. Moreover, we may restrict the distribution \(u\) to a subdomain \({\mathbb T}\times \Omega \) by considering it as a functional defined only on the test functions from \(\mathscr {S}(G)\) supported in \({\mathbb T}\times \Omega \).

### 2.6 Fourier transform

As a locally compact abelian group, \(G\) has a Fourier transform \(\mathscr {F}_{G}\) associated to it. The ability to utilize a Fourier transform that acts simultaneously in time \(t\in {\mathbb T}\) and space \(x\in \mathbb {R}^n\) shall play a key role in the following.

The projections introduced in (2.5) can be extended trivially to projections on the Schwartz–Bruhat space \({\mathcal P},{\mathcal P}_\bot :\mathscr {S}(G)\rightarrow \mathscr {S}(G)\). Introducing the delta distribution \(\delta _\mathbb {Z}\) on \(\frac{2\pi }{T}\mathbb {Z}\), that is, \(\delta _\mathbb {Z}(k):=1\) if \(k=0\) and \(\delta _\mathbb {Z}(k):=0\) for \(k\ne 0\), we observe that \({\mathcal P}u= \mathscr {F}^{-1}_G\big [\delta _\mathbb {Z}\mathscr {F}_G[u]\big ]\) and \({\mathcal P}_\bot u= \mathscr {F}^{-1}_G\big [(1-\delta _\mathbb {Z}) \mathscr {F}_G[u]\big ]\). Using these representations for \({\mathcal P}\) and \({\mathcal P}_\bot \), we naturally extend the projections to operators \({\mathcal P},{\mathcal P}_\bot :\mathscr {S^\prime }(G)\rightarrow \mathscr {S^\prime }(G)\). In accordance with the notation introduced above, we put \(\mathscr {S^\prime _\bot }(G):={\mathcal P}_\bot \mathscr {S^\prime }(G)\).

### Lemma 2.3

Let \(p\in (1,\infty )\) and Open image in new window . If Open image in new window for some parabolically 0-homogeneous \(\mathfrak {m}:\mathbb {R}\times \mathbb {R}^n\rightarrow \mathbb {C}\), then Open image in new window extends to a bounded operator Open image in new window .

### Proof

The Transference Principle (established originally by de Leeuw [10] and later extended to a general setting of locally compact abelian groups by Edwards and Gaudry [13, Theorem B.2.1]), makes it possible to “transfer” the investigation of Fourier multipliers from one group setting into another. In our case, [13, Theorem B.2.1] yields that \(\mathsf {m}\) is an \(L^{p}(G)\)-multiplier, provided \(\mathfrak {m}\) is an \(L^{p}(\mathbb {R}\times \mathbb {R}^n)\)-multiplier. To show the latter, we can employ one of the classical multiplier theorems available in the Euclidean setting. Since \(\mathfrak {m}\) is parabolically 0-homogeneous, it is easy to verify that \(\mathfrak {m}\) meets for instance the conditions of the Marcinkiewicz’s multiplier theorem ([24, Chapter IV, §6]). Thus, \(\mathfrak {m}\) is an \(L^{p}(\mathbb {R}\times \mathbb {R}^n)\)-multiplier and by [13, Theorem B.2.1] \(\mathsf {m}\) therefore an \(L^{p}(G)\)-multiplier. \(\square \)

### Lemma 2.4

Let \(p\in (1,\infty )\), Open image in new window and \(\alpha \le 0\). If Open image in new window for some parabolically \(\alpha \)-homogeneous function \(\mathfrak {m}:\mathbb {R}\times \mathbb {R}^n\setminus \{(0,0)\}\rightarrow \mathbb {C}\), then Open image in new window extends to a bounded operator Open image in new window .

### Proof

### Corollary 2.5

Let \(p\in (1,\infty )\) and \(\beta \in [0,1]\). If \(M\in C^{\infty }(\frac{2\pi }{T}\mathbb {Z}\setminus \{0\}\times \mathbb {R}^{n})\) is parabolically 0-homogeneous, then op [*M*] extends to a bounded operator op\([M]:W^{\beta ,2m\beta ,p}_{\bot }(G)\rightarrow W^{\beta ,2m\beta ,p}_{\bot }(G)\).

### 2.7 Time-periodic Bessel Potential spaces

### Proposition 2.6

Let \(p\in (1,\infty )\). Then \(H^{2m,p}_{\bot }(G)=W^{1,2m,p}_{\bot }(G)\) and \(H^{2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)=W^{1,2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\) with equivalent norms.

### Proof

It follows from Lemma 2.4 that \(\mathsf {op}\,\big [|\eta ,\xi |^{-m}\big ]\) extends to a bounded operator on \(L^{p}_\bot (G)\), which implies that \(||f||_{m,p}\) is equivalent to the norm \(||f||_p+||f||_{m,p}\). From this, we infer that \(H^{2m,p}_{\bot }=W^{1,2m,p}_{\bot }(G)\). A standard method (see for example [1, Theorem 4.26]) can be used to construct an extension operator \({{\mathrm{Ext}}}:W^{1,2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\rightarrow W^{1,2m,p}_{\bot }(G)\). The existence of an extension operator combined with the fact that \(H^{2m,p}_{\bot }=W^{1,2m,p}_{\bot }(G)\) implies \(H^{2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)=W^{1,2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\). \(\square \)

### Proposition 2.7

### Proof

### Lemma 2.8

Let \(\beta \in (0,1)\) and \(\alpha \in \big (2m(\beta -1),2m\beta \big )\). Then Open image in new window extends to a bounded operator Open image in new window .

### Proof

Finally, we characterize the trace spaces of the time-periodic Bessel Potential spaces.

### Lemma 2.9

### Proof

Consider now \(u\in H^{m,p}_{\bot }(G)\) with \({{\mathrm{supp}}}(u)\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\). As above we can identify \(u\) as an element of \(V(\mathbb {R})\), which necessarily satisfies \(u(0)=0\). It follows that \({{\mathrm{Tr}}}_m\big (u_{|{\mathbb T}\times \mathbb {R}^n_+}\big )=0\). If on the other hand \(u\in H^{m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\) with \({{\mathrm{Tr}}}_m\big (u_{|{\mathbb T}\times \mathbb {R}^n_+}\big )=0\), then it is standard to show that \(u\) can be approximated by a sequence of functions from \(C^{\infty }_0({\mathbb T}\times \mathbb {R}^n_+)\); see for example [26, Theorem 2.9.1]. Clearly, this sequence also converge in \(H^{m,p}_{\bot }(G)\). The limit function \(U\in H^{m,p}_{\bot }(G)\) satisfies \({{\mathrm{supp}}}U\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\) and \(U_{|{\mathbb T}\times \mathbb {R}^n_+}=u\). \(\square \)

## 3 Constant coefficients in the whole- and half-space

In this section, we establish the assertion of Theorem 1.4. We first treat the whole-space case, and then show the theorem as stated in the half-space case. Since we consider only the differential operators with constant coefficients in this section, we employ the simplified notation \(A(D)\) instead of \(A(x,D)\). Replacing the differential operator *D* with \(\xi \in \mathbb {R}^n\), we refer to \(A(\xi )\) as the symbol of \(A(D)\).

### 3.1 The whole space

We consider first the case of the spatial domain being the whole-space \(\mathbb {R}^n\). The time-space domain then coincides with the locally abelian group \(G\), and we can thus employ the Fourier transform \(\mathscr {F}_G\) and base the proof on an investigation of the corresponding Fourier multipliers.

### Lemma 3.1

### Proof

### Theorem 3.2

### Proof

### 3.2 The half space with Dirichlet boundary condition

In the next step, we consider the case of the spatial domain being the half-space \(\mathbb {R}^n_+\) and boundary operators corresponding to Dirichlet boundary conditions. As in the whole-space case, we shall work with the symbol of \(\partial _t+A^H\). In the following lemma, we collect its key properties.

### Lemma 3.3

- (1)
For every \((\eta ,\xi ')\in {\mathbb {R}\times \mathbb {R}^{n-1}\setminus \{(0,0)\}}\) the complex polynomial \(z\mapsto \mathfrak {M}(\eta ,\xi ',z)\) has exactly

*m*roots \(\rho _j^+(\eta ,\xi ')\in \mathbb {C}_-\) in the upper complex plane, and*m*roots \(\rho _j^-(\eta ,\xi ')\in \mathbb {C}_+\) in the lower complex plane \((j\in \{1,\ldots ,m\})\). - (2)The functionsare parabolically$$\begin{aligned}&\mathfrak {M}_\pm :\mathbb {R}\times \mathbb {R}^n\setminus \{(\eta ,\xi )\ |\ (\eta ,{\xi '})=(0,0)\} \rightarrow \mathbb {C},\nonumber \\&\mathfrak {M}_\pm (\eta ,\xi ):=\prod _{j=1}^{m}\big (\xi _n-\rho _j^\pm (\eta ,\xi ')\big ) \end{aligned}$$(3.4)
*m*-homogeneous. - (3)The coefficients of the polynomials \(z\mapsto \mathfrak {M}_\pm (\eta ,\xi ',z)\), more specifically the functions \(c^\pm _\alpha : \mathbb {R}\times \mathbb {R}^{n-1}\setminus \{(0,0)\}\rightarrow \mathbb {C}\)\((\alpha =0,\ldots ,m)\) with the property thatare analytic. Moreover, \(c^\pm _\alpha \) is parabolically \(\alpha \)-homogeneous.$$\begin{aligned} \mathfrak {M}_\pm (\eta ,\xi ',z) = \sum _{\alpha =0}^m c^\pm _\alpha (\eta ,{\xi '})z^{m-\alpha }, \end{aligned}$$(3.5)

### Proof

- (1)
Since \(A^H\) is properly elliptic, the polynomial \(z\mapsto \mathfrak {M}(0,\xi ',z)\) has exactly m roots in the upper and lower complex plane, respectively. Recall that \(A^H(x,\xi )\notin i\mathbb {R}\) for all \(\xi \in \mathbb {R}^n\setminus \{0\}\). Since the roots of a polynomial depend continuously on the polynomial’s coefficients, we deduce part (1) of the lemma.

- (2)
Since \(\mathfrak {M}\) is parabolically 2

*m*-homogeneous, the roots \(\rho _j^\pm \) are parabolically 1-homogeneous. It follows that \(\mathfrak {M}_\pm \) is parabolically*m*-homogeneous. - (3)
The analyticity of the coefficients \(c^\pm _\alpha \) follows by a classical argument; see for example [25, Chapter 4.4]. The coefficient \(c^\pm _\alpha \) being parabolically \(\alpha \)-homogeneous is a direct consequence of \(\mathfrak {M}_\pm \) being

*m*-homogeneous. \(\square \)

### Lemma 3.4

### Proof

*m*-homogeneous, we see that \(\mathfrak {M}_\pm \) can be bounded below by

The lemma above provides us with at decomposition of the differentiable operators in (3.2), that is, for \(\mathsf {A}:H^{s,p}_\bot (G)\rightarrow H^{s-2m,p}_\bot (G)\) and \(\mathsf {A}^{-1}:H^{s-2m,p}_\bot (G)\rightarrow H^{s,p}_\bot (G)\) the decompositions \(\mathsf {A}=\mathsf {A}_+\mathsf {A}_-=\mathsf {A}_-\mathsf {A}_+\) and \(\mathsf {A}^{-1}=\mathsf {A}_+^{-1}\mathsf {A}_-^{-1}=\mathsf {A}_-^{-1}\mathsf {A}_+^{-1}\) are valid provided \(\mathsf {A}\) is normalized accordingly. Employing the Paley–Wiener Theorem, we shall now show that the operators \(\mathsf {A}_\pm \) and \(\mathsf {A}^{-1}_\pm \) “respect” the support of a function in the upper (lower) half-space.

### Lemma 3.5

- (i)
If \({{\mathrm{supp}}}u\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\), then Open image in new window and Open image in new window .

- (ii)
If \({{\mathrm{supp}}}u\subset {\mathbb T}\times \overline{\mathbb {R}^n_-}\), then Open image in new window and Open image in new window .

### Proof

We shall prove only part (i), for part (ii) follows analogously. We employ the notation \({H}:={\mathbb T}\times \mathbb {R}^{n-1}\) and the canonical decomposition \(\mathscr {F}_G=\mathscr {F}_{H}\mathscr {F}_\mathbb {R}\) of the Fourier transform. In view of Lemma 3.4, it suffices to consider only \(u\in \mathscr {S}(G)\) with \({{\mathrm{supp}}}u\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\).

For fixed \({k}\in \frac{2\pi }{T}\mathbb {Z}\setminus \{0\}\) and \(\xi '\in \mathbb {R}^{n-1}\), we let \(\mathsf {D}({k},\xi '):=\mathscr {F}_{\mathbb {R}}^{-1}\mathsf {M}_+({k},\xi ',\cdot )\mathscr {F}_{\mathbb {R}}\). Since \(\mathsf {M}_+\) is a polynomial with respect to the variable \(\xi _n\), \(\mathsf {D}({k},\xi ')\) is a differential operator in \(x_n\) and hence \({{\mathrm{supp}}}(\mathsf {D}({k},\xi ')f)\subset \overline{\mathbb {R}_+}\) for every \(f\in \mathscr {S}'(\mathbb {R})\) with \({{\mathrm{supp}}}f\subset \overline{\mathbb {R}_+}\). Clearly, \({{\mathrm{supp}}}([\mathscr {F}_{H}u]({k},\xi ',\cdot ))\subset \overline{\mathbb {R}_+}\). Since \(\mathscr {F}_{H}[\mathsf {A}_+u]({k},\xi ',\cdot )=[\mathsf {D}({k},\xi ')\mathscr {F}_{H}u]({k},\xi ',\cdot )\), we conclude \({{\mathrm{supp}}}\mathsf {A}_+u\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\).

The above properties of \(\mathsf {A}_\pm \) and \(\mathsf {A}^{-1}_\pm \) lead to a surprisingly simple representation formula, see (3.8) below, for the solution \(u\) to the problem \(\partial _tu+A^Hu=f\) in the half-space \({\mathbb T}\times \mathbb {R}^n_+\) with Dirichlet boundary conditions. The problem itself can be formulated elegantly as (3.9).

### Lemma 3.6

### Proof

By Lemma 3.4, \(\mathsf {A}_-^{-1}f\in L^{p}_\bot (G)\). Clearly then \(Y_+\mathsf {A}_-^{-1}f\in L^{p}_\bot (G)\) and trivially \({{\mathrm{supp}}}(Y_+\mathsf {A}_-^{-1}f)\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\). Lemma 3.5 now implies \({{\mathrm{supp}}}(\mathsf {A}_+^{-1}Y_+\mathsf {A}_-^{-1}f)\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\), which concludes the first part of (3.9). Since \({{\mathrm{supp}}}\big ((Y_+-{{\mathrm{\textsf {id}}}})\mathsf {A}_-^{-1}f\big )\subset {\mathbb T}\times \overline{\mathbb {R}^n_-}\), Lemma 3.5 implies \({{\mathrm{supp}}}\big (\mathsf {A}_-(Y_+-{{\mathrm{\textsf {id}}}})\mathsf {A}_-^{-1}f\big )\subset {\mathbb T}\times \overline{\mathbb {R}^n_-}\). However, \(\mathsf {A}u-f=\mathsf {A}_-(Y_+-{{\mathrm{\textsf {id}}}})\mathsf {A}_-^{-1}f\), whence the second part of (3.9) follows.

To show uniqueness, let \(v\in H^{m,p}_{\bot }\) be a solution to (3.9) with \(f=0\). Since \(\mathsf {A}_+v=\mathsf {A}_-^{-1}\mathsf {A}v\), Lemma 3.5 yields \({{\mathrm{supp}}}(\mathsf {A}_+v)\subset {\mathbb T}\times \overline{\mathbb {R}^n_-}\). On the other hand, \({{\mathrm{supp}}}v\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\) by assumption, whence \({{\mathrm{supp}}}\mathsf {A}_+v\subset {\mathbb T}\times \overline{\mathbb {R}^n_+}\) by Lemma 3.5. Recalling that \(\mathsf {A}_+v\in L^{p}_\bot (G)\) by Lemma 3.4, we thus deduce \(\mathsf {A}_+v=0\) and consequently \(v=0\). This concludes the assertion of uniqueness.

Finally, we can establish the main theorem in the case of the spatial domain being the half-space \(\mathbb {R}^n_+\) and boundary operators corresponding to Dirichlet boundary conditions.

### Theorem 3.7

### Proof

*m*steps the desired regularity \(u\in H^{2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\) together with the estimate \(||u||_{2m,p,{\mathbb T}\times \mathbb {R}^n_+}\le ||f||_p\). Recalling from Proposition 2.6 that \(H^{2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)=W^{1,2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\), we conclude (3.12) in the case \(g=0\).

To show uniqueness, assume \(u\in W^{1,2m,p}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\) is a solution the (3.11) with homogeneous data \(f=g=0\). By Lemma 2.9 there is an extension \(U\in W^{1,2m,p}_{\bot }(G)\) of \(u\) with \({{\mathrm{supp}}}U\in {\mathbb T}\times \overline{\mathbb {R}^n_+}\). By Lemma 3.6, \(U=0\). \(\square \)

### 3.3 The half space with general boundary conditions

Let \(\Omega :=\mathbb {R}^n_+\) and \((A,B_1,\ldots ,B_m)\) be differential operators of the form (1.4) with constant coefficients.

### Lemma 3.8

*z*the mappings Open image in new window and \(z\mapsto B^H_j(\xi ',z)\), where Open image in new window is defined as in Lemma 3.4. For \(j=1,\ldots ,m\) let \(F_{(j-1)l}(k,\xi ')\), \(l=0,\ldots ,m-1\), denote the coefficients of the polynomial Open image in new window . The corresponding matrix \(F(k,\xi ')\in \mathbb {C}^{m\times m}\) is called

*characteristic matrix*. The characteristic matrix function \(F:\frac{2\pi }{T}\mathbb {Z}\times \mathbb {R}^{n-1}\setminus \{(0,0)\}\rightarrow \mathbb {C}^{m\times m}\) has an extension \(F:\mathbb {R}\times \mathbb {R}^{n-1}\setminus \{(0,0)\}\rightarrow \mathbb {C}^{m\times m}\) that satisfies \((j,l=0,\ldots ,m-1)\)

### Proof

*j*’th row of the characteristic matrix \(F\) to \((\eta ,{\xi '})\in \mathbb {R}\times \mathbb {R}^{n-1}\setminus \{(0,0)\}\). By definition we have

*j*and

*l*. Since \(F^{-1}(\eta ,{\xi '})=\big (\det F(\eta ,{\xi '})\big )^{-1}{{\mathrm{cof}}}F(\eta ,{\xi '})^\top \), (3.17) follows from (3.14). Finally, (3.18) follows from (3.16) in the same way (3.15) was derived from (3.13). \(\square \)

### Lemma 3.9

### Proof

### Lemma 3.10

### Proof

*R*and letting \(R\rightarrow \infty \) (see for example [19, Chapter 2, Proposition 4.1]), (3.23) follows. \(\square \)

### Lemma 3.11

Let the assumptions be as in Lemma 3.8. If \(u\in W^{1,2m,2}_{\bot }({\mathbb T}\times \mathbb {R}^n_+)\) satisfies Open image in new window , then Open image in new window .

### Proof

Employ the partial Fourier transform \(\mathscr {F}_{{{\mathbb T}\times \mathbb {R}^{n-1}}}\) to the equation \(\mathsf {A}u=0\), which in view of Plancherel’s theorem implies \(\mathsf {A}(k,\xi ',D_n)\mathscr {F}_{{{\mathbb T}\times \mathbb {R}^{n-1}}}(u)=0\) for almost every \((k,\xi ')\). By Lemma 3.10, \(B^H_j(\xi ',D_n)\mathscr {F}_{{{\mathbb T}\times \mathbb {R}^{n-1}}}(u)={F_{(j-1)l}(k,\xi ')\big ({{\mathrm{Tr}}}_m u(0)\big )_l}\). Employing \(\mathscr {F}^{-1}_{{{\mathbb T}\times \mathbb {R}^{n-1}}}\), we obtain \(B^Hu= \mathsf {op}\,[F]{{\mathrm{Tr}}}_mu\). \(\square \)

### Theorem 3.12

### Proof

## 4 Proof of the main theorems

### Proof of Theorem 1.4

As already noted in Sect. 2.4, the canonical bijection between \(C^{\infty }_0({\mathbb T}\times \mathbb {R}^n_+)\) and \(C^{\infty }_{0,\mathrm {per}}(\mathbb {R}\times \mathbb {R}^n_+)\) implies that \(W^{1,2m,p}\big ({\mathbb T}\times \mathbb {R}^n_+\big )\) and \(W^{1,2m,p}_{\mathrm {per}}(\mathbb {R}\times \mathbb {R}^n_+)\) are isometrically isomorphic. It follows that \(W^{1,2m,p}_{\bot }\big ({\mathbb T}\times \mathbb {R}^n_+\big )\) and \({\mathcal P}_\bot W^{1,2m,p}_{\mathrm {per}}(\mathbb {R}\times \mathbb {R}^n_+)\) as well as \(T_\bot ^{\kappa ,p}({\mathbb T}\times \partial \mathbb {R}^n_+)\) and \(\Pi _{j=1}^m {\mathcal P}_\bot W^{\kappa _j,2m\kappa _j,p}_{\mathrm {per}}(\mathbb {R}\times \partial \mathbb {R}^n_+)\) are isometrically isomorphic. Consequently, Theorem 1.4 follows from Theorem 3.12. \(\square \)

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society.

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