Boundary value problems and Hardy spaces for elliptic systems with block structure

Abstract

For elliptic systems with block structure in the upper half-space and t-independent coefficients, we settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal ranges of exponents. Prior to this work, only the two-dimensional situation was fully understood. In higher dimensions, partial results for existence in smaller ranges of exponents and for a subclass of such systems had been established. The presented uniqueness results are completely new. We also elucidate optimal ranges for problems with fractional regularity data. Methods use and improve, with some new results, all the machinery developed over the last two decades to study such problems: the Kato square root estimates and Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates, non-tangential estimates and square functions and abstract layer potentials to replace fundamental solutions in the absence of local regularity of solutions. This self-contained monograph provides a comprehensive overview on the field and unifies many earlier results that have been obtained by a variety of methods.

Figures (18)

• Figure 1: Compatible well-posedness region for Besov and Hardy--Sobolev data when $p_+(L) \leq n$.
• Figure 2: Compatible well-posedness region for Besov and Hardy--Sobolev data when $p_+(L) > n$ but $p_-(L^\sharp) \geq 1$.
• Figure 3: Compatible well-posedness region for Besov and Hardy--Sobolev data when $p_-(L^\sharp) < 1$. This implies $p_+(L) = \infty$. Hence, at the bottom there is no horizontal thick red line as in Figure \ref{['fig: diagram p+large']} for $1/p > 0$ and Theorem \ref{['thm: Holder-dir']} yields compatible well-posedness for the full horizontal segment with $1/p \leq 0$. The number $\theta$ comes from the first-order approach in AA. It has a specific meaning, see Proposition \ref{['prop: blue region large']}, and is not larger than $n(1/p_-(L^\sharp) -1)$, which is the limitation of part (iii) in Theorem \ref{['thm: Holder-dir']} for $\dot{\Lambda}{\newline}^\alpha$-data. When $1-1/p_-(L^\sharp) < -\alpha/n \leq 0$, well-posedness of the $\dot{\Lambda}{\newline}^\alpha$ Dirichlet problem for $\mathcal{L}$ can also be obtained by duality from well-posedness of the $\operatorname{H}^{n/(\alpha+n)}$ regularity problem for $\mathcal{L}^*$, using $DB^*$-adapted spaces AA. This is why the corresponding horizontal segment for $1/p = -\alpha/n \leq 0$ has been colored in gray.
• Figure 4: Visualization of the proof of Lemma \ref{['lem: extra']}.
• Figure 5: Canonical completion: $\varphi, \psi \in \Psi_\infty^\infty$ are siblings and $P$ is the unique bounded linear map for which the diagram commutes. It follows that $P$ is a projection from $\operatorname{Y}^{s,p}$ onto $\psi \mathbb{X}^{s,p}$. By the universal approximation technique for $\operatorname{Y}$-spaces, projections for different choices of admissible spaces are compatible. The bottom part of the diagram also identifies $\psi \mathbb{X}^{s,p} \cap \mathbb{Q}_{\psi,T}(\mathbb{X}^{0,2}_T) = \mathbb{Q}_{\psi,T}(\mathbb{X}^{s,p}_T$).

Theorems & Definitions (351)

• Theorem 1.1: Dirichlet problem
• Theorem 1.2: Regularity problem
• Theorem 1.3: $\dot{\Lambda}{\newline}^\alpha$ Dirichlet problem
• Theorem 1.4
• Theorem 1.5: Neumann problem
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5