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Global Hölder solvability of linear and quasilinear Poisson equations

Takanobu Hara

TL;DR

The paper addresses global Hölder solvability for linear and quasilinear Poisson-type equations with measure data on bounded domains. It develops a barrier-based approach under a capacity density condition and Morrey-type measure data, yielding a unique weak solution that is globally Hölder continuous with explicit exponents and boundary control. A key contribution is a constructive method for supersolutions and a criterion linking solvability to the Morrey-type boundary measures, plus a compact embedding theorem for W^{1,p}_0(Ω) into L^{p}(Ω;ν). The results extend classical Poisson theory to Lipschitz domains with minimal data assumptions and offer a unified framework connecting capacity, measure data, and global boundary regularity with embedding consequences.

Abstract

We establish an existence result for globally continuous weak solutions to elliptic equations of the $p$-Poisson type. This result significantly improves Theorem 8.30 in Gilbarg-Trudinger (1983) and offers a novel contribution for the classical Poisson equation on Lipschitz domains, ensuring global Hölder continuity of solutions under a minimal assumption on the right-hand side. Applications of this result to embedding theorems are also discussed.

Global Hölder solvability of linear and quasilinear Poisson equations

TL;DR

The paper addresses global Hölder solvability for linear and quasilinear Poisson-type equations with measure data on bounded domains. It develops a barrier-based approach under a capacity density condition and Morrey-type measure data, yielding a unique weak solution that is globally Hölder continuous with explicit exponents and boundary control. A key contribution is a constructive method for supersolutions and a criterion linking solvability to the Morrey-type boundary measures, plus a compact embedding theorem for W^{1,p}_0(Ω) into L^{p}(Ω;ν). The results extend classical Poisson theory to Lipschitz domains with minimal data assumptions and offer a unified framework connecting capacity, measure data, and global boundary regularity with embedding consequences.

Abstract

We establish an existence result for globally continuous weak solutions to elliptic equations of the -Poisson type. This result significantly improves Theorem 8.30 in Gilbarg-Trudinger (1983) and offers a novel contribution for the classical Poisson equation on Lipschitz domains, ensuring global Hölder continuity of solutions under a minimal assumption on the right-hand side. Applications of this result to embedding theorems are also discussed.
Paper Structure (9 sections, 18 theorems, 89 equations)

This paper contains 9 sections, 18 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background and contribution of the paper
  4. Compare with known results
  5. Preliminaries
  6. Construction of supersolutions
  7. Existence of globally Hölder continuous solutions
  8. Applications to embedding theorems
  9. Examples

Key Result

Theorem 1.1

Assume that $\Omega$ satisfies the capacity density condition Let $\nu$ be a locally finite signed Radon measure satisfying where $M$ is a constant, $n - p < \lambda \le n$, and $\delta(x) := \mathrm{dist}(x, \partial \Omega)$. Then, there exists a unique weak solution $u \in W^{1, p}_{\mathrm{loc}}(\Omega) \cap C(\overline{\Omega})$ to eqn:DE. Moreover, $u \in C^{\beta_{1}}( \overline{\Omega}

Theorems & Definitions (38)

  • Theorem 1.1
  • Example 1.2
  • Example 1.3
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 28 more