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Tangential approach in the Dirichlet problem for elliptic equations

Jonathan Bennett, Arnaud Dumont, Andrew J. Morris

TL;DR

This work addresses the boundary behavior of solutions to divergence-form elliptic equations with measurable, uniformly elliptic coefficients on Lipschitz domains, linking the solvability of the $L^p$ Dirichlet problem to quantitative absolute continuity of the $L$-harmonic measure via $A_{\infty}$-type conditions. The authors develop a robust tangential-convergence theory: for data in Bessel potential spaces $\mathscr{L}^{p}_{\alpha}$ (and, by extension, $\mathrm{C}^{p}_{\alpha}$ and related classes), tangential boundary limits along regions $\Gamma^{\beta}$ hold a.e. and the nontangential maximal control can be upgraded to tangential maximal control, with sharp Hausdorff-dimension bounds on the divergence set. The results are proved first in the upper half-space using Frostman-type arguments and Bessel-potential machinery, then extended to bounded Lipschitz domains via localization, corkscrew geometry, and boundary-square function estimates, culminating in a comprehensive tangential-convergence theory for Sobolev boundary data. The findings extend Nagel–Stein–Rudin–Shapiro and Dorronsoro-type results to general Lipschitz domains, providing precise tradeoffs between smoothness, domain geometry, and boundary convergence behavior with clear implications for PDE boundary-value problems.

Abstract

It is well-known that solvability of the $\mathrm{L}^{p}$-Dirichlet problem for elliptic equations $Lu:=-\mathrm{div}(A\nabla u)=0$ with real-valued, bounded and measurable coefficients $A$ on Lipschitz domains $Ω\subset\mathbb{R}^{1+n}$ is characterised by a quantitative absolute continuity of the associated $L$-harmonic measure. We prove that this local $A_{\infty}$ property is sufficient to guarantee that the nontangential convergence afforded to $\mathrm{L}^{p}$ boundary data actually improves to a certain \emph{tangential} convergence when the data has additional (Sobolev) regularity. Moreover, we obtain sharp estimates on the Hausdorff dimension of the set on which such convergence can fail. This extends results obtained by Dorronsoro, Nagel, Rudin, Shapiro and Stein for classical harmonic functions in the upper half-space.

Tangential approach in the Dirichlet problem for elliptic equations

TL;DR

This work addresses the boundary behavior of solutions to divergence-form elliptic equations with measurable, uniformly elliptic coefficients on Lipschitz domains, linking the solvability of the Dirichlet problem to quantitative absolute continuity of the -harmonic measure via -type conditions. The authors develop a robust tangential-convergence theory: for data in Bessel potential spaces (and, by extension, and related classes), tangential boundary limits along regions hold a.e. and the nontangential maximal control can be upgraded to tangential maximal control, with sharp Hausdorff-dimension bounds on the divergence set. The results are proved first in the upper half-space using Frostman-type arguments and Bessel-potential machinery, then extended to bounded Lipschitz domains via localization, corkscrew geometry, and boundary-square function estimates, culminating in a comprehensive tangential-convergence theory for Sobolev boundary data. The findings extend Nagel–Stein–Rudin–Shapiro and Dorronsoro-type results to general Lipschitz domains, providing precise tradeoffs between smoothness, domain geometry, and boundary convergence behavior with clear implications for PDE boundary-value problems.

Abstract

It is well-known that solvability of the -Dirichlet problem for elliptic equations with real-valued, bounded and measurable coefficients on Lipschitz domains is characterised by a quantitative absolute continuity of the associated -harmonic measure. We prove that this local property is sufficient to guarantee that the nontangential convergence afforded to boundary data actually improves to a certain \emph{tangential} convergence when the data has additional (Sobolev) regularity. Moreover, we obtain sharp estimates on the Hausdorff dimension of the set on which such convergence can fail. This extends results obtained by Dorronsoro, Nagel, Rudin, Shapiro and Stein for classical harmonic functions in the upper half-space.

Paper Structure

This paper contains 22 sections, 37 theorems, 242 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background: Laplace's equation in the upper half-space
  3. The Dirichlet problem for elliptic equations
  4. Preliminaries
  5. Notation
  6. Sobolev spaces
  7. Fractional dimensional measures and Frostman's lemma
  8. The Dirichlet problem in Hn and the proof of Theorem \ref{['main theorem upper half space']}
  9. Nontangential approach in the Lp-Dirichlet problem in Hn
  10. Further properties of Bessel potentials
  11. Tangential maximal functions
  12. Further properties of the spaces Cpalpha and an extension of Theorem \ref{['main theorem upper half space']}
  13. The Dirichlet problem in bounded Lipschitz domains and the proof of Theorem \ref{['second main theorem bounded Lipschitz domain']}
  14. Lipschitz domains
  15. The L-harmonic measure in Lipschitz domains
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\gamma\subset\mathbb{R}^{1+n}_{+}$ be a tangential curve such that $(0,0)\in\overline{\gamma}$. There exists $f\in\mathrm{L}^{\infty}(\mathbb{R}^{n})$ with Poisson extension $u_{f}(t,x)=(P_{t}\ast f)(x)$ such that the limit does not exist for any $x_0\in\mathbb{R}^{n}$.

Figures (1)

  • Figure 1: A part of the boundary $\partial\Omega$ of a Lipschitz domain $\Omega$ and some tangential approach regions ($n=1$, $\beta=\tfrac{1}{2}$, $c=1$).

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2: Frostman's lemma
  • ...and 56 more