The Generalized Hölder and Morrey-Campanato Dirichlet Problems for Elliptic Systems in the Upper Half-Space

Abstract

We prove well-posedness results for the Dirichlet problem in $\mathbb {R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces $\mathscr{C}^{\omega }(\mathbb {R}^{n-1},\mathbb {C}^{M})$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{\omega ,p} (\mathbb {R}^{n-1},\mathbb {C}^{M})$ under certain assumptions on the growth function ω. We also identify a class of growth functions ω for which $\mathscr{C}^{\omega }(\mathbb {R}^{n-1},\mathbb {C}^{M})=\mathscr{E}^{\omega ,p}(\mathbb {R}^{n-1},\mathbb {C}^{M})$ and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates.

Appendix: John-Nirenberg’s Inequality Adapted to Growth Functions

In what follows we assume that all cubes are half-open, that is, they can be written in the form $Q=[a_{1},a_{1}+\ell (Q))\times \dots \times [a_{n-1},a_{n-1}+\ell (Q))$ with $a_{i}\in \mathbb {R}^{n-1}$ and ℓ(Q) > 0. Notice that since ∂Q has Lebesgue measure zero the said assumption is harmless. Subdividing dydically yields the collection of (half-open) dyadic-subcubes of a given cube Q, which we shall denote by $\mathbb {D}_{Q}$. For the following statement, and with the aim of considering global results, it is also convenient to consider the case $Q=\mathbb {R}^{n-1}$ in which scenario we take $\mathbb {D}_{Q}$ to be the classical dyadic grid generated by [0,1)^n− 1, or any other dyadic grid. Let us also recall the definition of the dyadic Hardy-Littlewood maximal function localized to a given cube Q, i.e.,

(A.1)

for eachf ∈ L¹(Q). The following result is an extension of the John-Nirenberg inequality obtained in [3, 4] (when φ ≡ 1) adapted to our growth function.

Lemma A.1

Let $F\in L^{2}_{\textup {loc}}(\mathbb {R}^{n}_{+})$and let $\varphi :[0,\infty )\to [0,\infty )$be a non-decreasing function. Let $Q_{0}\subset \mathbb {R}^{n-1}$be an arbitrary half-open cube, or $Q_{0}=\mathbb {R}^{n-1}$. Assume that there are numbers α ∈ (0, 1) and $N\in (0,\infty )$such that

$$ \bigg|\left\{x^{\prime}\in Q: \frac{1}{\varphi(\ell(Q))}\left( {\int}_{0}^{\ell(Q)}{\int}_{|x^{\prime}-y^{\prime}|<\kappa s} |F(y^{\prime},s)|^{2} dy^{\prime}\frac{ds}{s^{n}}\right)^{1/2}>N\right\}\bigg|\leq\alpha|Q| $$

(A.2)

for every cube $Q\in \mathbb {D}_{Q_{0}}$and $\kappa := 1+2\sqrt {n-1}$. Then, for every t > 0

$$ \begin{array}{@{}rcl@{}} \sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\frac{1}{|Q|}\bigg|\left\{x^{\prime}\in Q: \frac{1}{\varphi(\ell(Q))} \left( {\int}_{0}^{\ell(Q)}{\int}_{|x^{\prime}-y^{\prime}|<\kappa s}|F(y^{\prime},s)|^{2} dy^{\prime}\frac{ds}{s^{n}}\right)^{1/2}>t\right\}\bigg| \\ \leq\frac{1}{\alpha} e^{-\tfrac{\log(\alpha^{-1})}{N} t}. \end{array} $$

(A.3)

Hence, for each $q\in (0,\infty )$there exists a finite constant C = C(α, q) ≥ 1 such that

(A.4)

Moreover, there exists some finite C = C(α) ≥ 1 such that

$$ \sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\frac{1}{\varphi(\ell(Q))} \bigg\|\left( {\int}_{0}^{\ell(Q)}{\int}_{| \cdot -y^{\prime}|<s}|F(y^{\prime},s)|^{2} dy^{\prime}\frac{ds}{s^{n}}\right)^{1/2}\bigg\|_{\exp L,Q}\leq CN. $$

(A.5)

The previous result can be proved using the arguments in [3, 4] with appropriate modifications. Here we present an alternative abstract argument based on ideas that go back to Calderón, as presented in [11] (see also [5, 12]). This also contains as a particular case the classical John-Nirenberg result concerning the exponential integrability of BMO functions.

Proposition A.2

Let $Q_{0}\subset \mathbb {R}^{n-1}$be an arbitrary half-open cube, or $Q_{0}=\mathbb {R}^{n-1}$. For every $Q\in \mathbb {D}_{Q_{0}}$assume that there exist two non-negative functions $G_{Q}, H_{Q}\in L^{1}_{\textup {loc}}(\mathbb {R}^{n-1})$such that

$$ G_{Q}(x^{\prime})\leq H_{Q}(x^{\prime}) \text{ for almost every } x^{\prime}\in Q, $$

(A.6)

and, for every $Q^{\prime }\in \mathbb {D}_{Q}\setminus \{Q\}$,

$$ G_{Q}(x^{\prime})\leq G_{Q^{\prime}}(x^{\prime})+H_{Q}(y^{\prime}) \text{ for a.e.}~ x\in Q^{\prime} ~\text{and for a.e. } y^{\prime}\in\widehat{Q^{\prime}}, $$

(A.7)

where $\widehat {Q^{\prime }}$is the dyadic parent of $Q^{\prime }$. For each α ∈ (0, 1) define

$$ \mathfrak{m}_{\alpha}:=\sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\operatornamewithlimits{inf\vphantom{p}}\left\{\lambda>0: |\{x^{\prime}\in Q: H_{Q}(x^{\prime})>\lambda\}| \leq\alpha|Q|\right\}. $$

(A.8)

Then, for every α ∈ (0, 1) one has

$$ \sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\frac{|\{x\in Q: G_{Q}(x^{\prime})>t\}|}{|Q|} \leq\frac{1}{\alpha} e^{-\log(\alpha^{-1}) \tfrac{t}{\mathfrak{m}_{\alpha}}},\qquad\forall t>0. $$

(A.9)

As a consequence,

$$ \sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\|G_{Q}\|_{\exp L,Q}\leq\frac{1+\alpha^{-1}}{\log (\alpha^{-1})} \mathfrak{m}_{\alpha} $$

(A.10)

and for every $q\in (0,\infty )$there exists a finite constant C = C(q) > 0 such that

(A.11)

Before proving this result and Lemma A.1, let us illustrate how Proposition A.2 yields the classical John-Nirenberg result regarding the exponential integrability of BMO functions. Concretely, pick $f\in \text {BMO}(\mathbb {R}^{n-1})$. Fix an arbitrary cube Q₀ and for every $Q\in \mathbb {D}_{Q_{0}}$ define G_Q := |f −f_Q| and $H_{Q}:=2^{n-1}{M^{d}_{Q}}\left (|f-f_{Q}|\right )$ (cf. Eq. A.1). Clearly, Eq. A.6 holds by Lebesgue’s Differentiation Theorem. Moreover, for every $Q^{\prime }\in \mathbb {D}_{Q}\setminus Q$, $x^{\prime }\in Q^{\prime }$, and $y^{\prime }\in \widehat {Q^{\prime }}$ we have

(A.12)

and Eq. A.7 follows. Going further, by the weak-type(1, 1) of the dyadic Hardy-Littlewood maximal function, for every λ > 0 we may write

$$ |\{x^{\prime}\in Q: H_{Q}(x^{\prime})>\lambda\}|\leq\frac{2^{n-1}}{\lambda}{\int}_{Q}|f(y^{\prime})-f_{Q}| dy^{\prime} \leq\frac{2^{n-1}\|f\|_{\text{BMO}(\mathbb{R}^{n-1})}}{\lambda}|Q|. $$

(A.13)

In particular, choosing for instance α := e^− 1, if we use the previous estimate with $\lambda :=2^{n-1}\|f\|_{\text {BMO}(\mathbb {R}^{n-1})}/\alpha$ we obtain $\mathfrak {m}_{\alpha }\le 2^{n-1}\|f\|_{\text {BMO}(\mathbb {R}^{n-1})}/\alpha$. Thus, Eq. A.9 yields

$$ \begin{array}{@{}rcl@{}} \frac{|\{x\in Q_{0}: |f(x^{\prime})-f_{Q_{0}}|>t\}|}{|Q_{0}|} &\leq&\frac1\alpha e^{-\tfrac{\alpha\log(\alpha^{-1})}{2^{n-1}}\tfrac{t}{\|f\|_{\text{BMO}(\mathbb{R}^{n-1})}}} \\ &=&e\cdot e^{-\tfrac1 {2^{n-1}e}\tfrac{t}{\|f\|_{\text{BMO}(\mathbb{R}^{n-1})}}} \end{array} $$

(A.14)

while Eq. A.10 gives

$$ \|f-f_{Q_{0}}\|_{\exp L,Q_{0}}\leq(1+e) e 2^{n-1}\|f\|_{\text{BMO}(\mathbb{R}^{n-1})} $$

(A.15)

which are the well-known John-Nirenberg inequalities.

We now turn to the proof of Lemma A.1.

Proof of Lemma A.1

Let F, α, and N be fixed as in the statement of the lemma. For every $Q\in \mathbb {D}_{Q_{0}}$ and $x^{\prime } \in \mathbb {R}^{n-1}$, define

$$ G_{Q}(x^{\prime}):=\frac{1}{\varphi(\ell(Q))} \left( {\int}_{0}^{\ell(Q)} {\int}_{|x^{\prime}-z^{\prime}|<s}|F(z^{\prime},s)|^{2} dz^{\prime}\frac{ds}{s^{n}} \right)^{1/2} $$

(A.16)

and

$$ H_{Q}(x^{\prime})=\frac{1}{\varphi(\ell(Q))} \left( {\int}_{0}^{\ell(Q)}{\int}_{|x^{\prime}-z^{\prime}|<\kappa s}|F(z^{\prime},s)|^{2} dz^{\prime}\frac{ds}{s^{n}} \right)^{1/2}. $$

(A.17)

Note that (A.6) is trivially verified since κ > 1. To proceed, fix $Q^{\prime }\in \mathbb {D}_{Q}$ along with $x^{\prime }\in Q^{\prime }$ and $y^{\prime }\in \widehat {Q^{\prime }}$. If $|x^{\prime }-z^{\prime }|<s$ with $\ell (Q^{\prime })\leq s\le \ell (Q)$ then

$$ |y^{\prime}-z^{\prime}|\leq|y^{\prime}-x^{\prime}|+|x^{\prime}-z^{\prime}|<2\sqrt{n-1} \ell(Q^{\prime})+s\leq\kappa s. $$

(A.18)

Therefore, since φ is non-decreasing,

$$ \begin{array}{@{}rcl@{}} G_{Q}(x^{\prime})\leq\frac{\varphi(\ell(Q^{\prime}))}{\varphi(\ell(Q))}G_{Q^{\prime}}(x^{\prime}) +\frac{1}{\varphi(\ell(Q))} \left( {\int}_{\ell(Q^{\prime})}^{\ell(Q)}{\int}_{|x^{\prime}-z^{\prime}|<s}|F(z^{\prime},s)|^{2} dz^{\prime}\frac{ds}{s^{n}} \right)^{1/2} \\ \leq G_{Q^{\prime}}(x^{\prime})+H_{Q}(y^{\prime}), \end{array} $$

(A.19)

establishing (A.7). Moreover, Eq. A.2 gives immediately that $\mathfrak {m}_{\alpha }\le N$. Granted this, Eqs. A.9, A.11, and A.10, (with α ∈ (0, 1) given by Eq. A.2) prove, respectively Eqs. A.3, A.4, and A.5. □

Finally, we give the proof of Proposition A.2.

Proof of Proposition A.2

We start by introducing some notation. Set

$$ {\Xi}(t):=\sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\frac{|E_{Q}(t)|}{|Q|} :=\sup\limits_{Q\in\mathbb{D}_{Q_{0}}}\frac{|\{x^{\prime}\in Q: G_{Q}(x^{\prime})>t\}|}{|Q|},\qquad 0<t<\infty. $$

(A.20)

Fix α ∈ (0,1), let ε > 0 be arbitrary, and write $\lambda _{\varepsilon }=\mathfrak {m}_{\alpha }+\varepsilon$. From Eq. A.8 it follows that

$$ |F_{Q,\varepsilon}|:=|\{x^{\prime}\in Q: H_{Q}(x^{\prime})>\lambda_{\varepsilon}\}|\leq\alpha|Q|, \qquad\forall Q\in\mathbb{D}_{Q_{0}}. $$

(A.21)

Fix now $Q\in \mathbb {D}_{Q_{0}}$, β ∈ (α,1) (we will eventually let β → 1⁺) and set

$$ {\Omega}_{Q}:=\{x^{\prime}\in Q: {M_{Q}^{d}}(1_{F_{Q,\varepsilon}})(x^{\prime})>\beta\}. $$

(A.22)

Note that (A.21) ensures that

(A.23)

hence we can extract a family of pairwise disjoint stopping-time cubes $\{Q_{j}\}_{j}\subset \mathbb {D}_{Q}\setminus \{Q\}$ so that Ω_Q = ∪_jQ_j and for every j

$$ \frac{|F_{Q,\varepsilon}\cap Q_{j}|}{|Q_{j}|}>\beta, \quad\frac{|F_{Q,\varepsilon}\cap Q^{\prime}|}{|Q^{\prime}|}\leq\beta, \quad Q_{j}\subsetneq Q^{\prime}\in\mathbb{D}_{Q}. $$

(A.24)

Let t > λ_ε and note that (A.6) gives

$$ \lambda_{\varepsilon}<t<G_{Q}(x^{\prime})\leq H_{Q}(x^{\prime}) \text{ for a.e. } x^{\prime}\in E_{Q}(t). $$

(A.25)

which implies that

$$ \beta<1={\mathbf{1}}_{F_{Q,\varepsilon}}(x^{\prime})\leq {M_{Q}^{d}}\left( {\mathbf{1}}_{F_{Q,\varepsilon}}\right)(x^{\prime}) \text{ for a.e. } x^{\prime}\in E_{Q}(t). $$

(A.26)

Hence,

$$ |E_{Q}(t)|=|E_{Q}(t)\cap{\Omega}_{Q}|=\sum\limits_{j} |E_{Q}(t)\cap Q_{j}|. $$

(A.27)

For every j, by the second estimate in Eq. A.24 applied to $\widehat {Q}_{j}$, the dyadic parent of Q_j, we have $|F_{Q,\varepsilon }\cap \widehat {Q}_{j}|/|\widehat {Q}_{j}|\leq \beta <1$, therefore $|\widehat {Q}_{j}\setminus F_{Q,\varepsilon }|/|\widehat {Q}_{j}|>1-\beta >0$. In particular, Eq. A.7 guarantees that we can find $\widehat {x}_{j}^{\prime }\in \widehat {Q}_{j}\setminus F_{Q,\varepsilon }$, such that for a.e. $x^{\prime }\in Q_{j}$ we have

$$ \begin{array}{@{}rcl@{}} G_{Q}(x^{\prime})\leq G_{Q_{j}}(x^{\prime})+ H_{Q}(\widehat{x}_{j}^{\prime})\leq G_{Q_{j}}(x^{\prime})+\lambda_{\varepsilon}. \end{array} $$

(A.28)

Consequently, $G_{Q_{j}}(x^{\prime })>t-\lambda _{\varepsilon }$ for a.e. $x^{\prime }\in E_{Q}(t)\cap Q_{j}$ which further implies

$$ |E_{Q}(t)\cap Q_{j}|\leq|\{x^{\prime}\in Q_{j}: G_{Q_{j}}(x^{\prime})>t-\lambda_{\varepsilon}\}| \leq{\Xi}(t-\lambda_{\varepsilon})|Q_{j}|. $$

(A.29)

In turn, this permits us to estimate

$$ \begin{array}{@{}rcl@{}} |E_{Q}(t)|=\sum\limits_{j}|E_{Q}(t)\cap Q_{j}|\leq{\Xi}(t-\lambda_{\varepsilon})\sum\limits_{j}|Q_{j}| \leq{\Xi}(t-\lambda_{\varepsilon})\frac1{\beta}\sum\limits_{j} |F_{Q,\varepsilon}\cap Q_{j}| \\ \leq{\Xi}(t-\lambda_{\varepsilon})\frac1\beta|F_{Q,\varepsilon}| \leq{\Xi}(t-\lambda_{\varepsilon})\frac\alpha\beta|Q|, \end{array} $$

(A.30)

where we have used (A.24), that the cubes {Q_j}_j are pairwise disjoint and, finally, Eq. A.21. Dividing by |Q| and taking the supremum over all $Q\in \mathbb {D}_{Q_{0}}$ we arrive at

$$ {\Xi}(t)\leq\frac\alpha\beta{\Xi}(t-\lambda_{\varepsilon}),\qquad t>\lambda_{\varepsilon}. $$

(A.31)

Since this is valid for all β ∈ (α,1), we can now let β → 1⁺, iterate the previous expression, and use the fact that Ξ(t) ≤ 1 to conclude that

$$ {\Xi}(t)\leq\frac1{\alpha}\alpha^{\frac{t}{\lambda_{\varepsilon}}} =\frac{1}{\alpha} e^{-\log(\alpha^{-1}) \frac{t}{\lambda_{\varepsilon}}},\qquad t>0. $$

(A.32)

Recalling that $\lambda _{\varepsilon }=\mathfrak {m}_{\alpha }+\varepsilon$ and letting ε → 0⁺ establishes (A.7).

We shall next indicate how (A.9) implies (A.10). Concretely, if we take $t:=\frac {1+\alpha ^{-1}}{\log (\alpha ^{-1})} \mathfrak {m}_{\alpha }$ we see that (A.9) gives

(A.33)

With this in hand, Eq. A.10follows with the help of Eq. 1.16.

At this stage, there remains to justify (A.11). This can be done invoking again (A.9):

(A.34)

This completes the proof of Proposition A.2. □