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Uniformly Elliptic Equations on Domains with Capacity Density Conditions: Existence of Hölder Continuous Solutions and Homogenization Results

Takanobu Hara

Abstract

This is a progress report on study of uniformly elliptic Poisson-type equations on domains with capacity density conditions (CDC domains). We give a brief summary of known facts of CDC domains, including Hardy's inequality, and review a previous work of existence of globally Hölder continuous solutions. Additionally, we apply the result to homogenization problems of $ε$-periodic coefficients and present a convergence rate estimate of $L^{\infty}$ norms.

Uniformly Elliptic Equations on Domains with Capacity Density Conditions: Existence of Hölder Continuous Solutions and Homogenization Results

Abstract

This is a progress report on study of uniformly elliptic Poisson-type equations on domains with capacity density conditions (CDC domains). We give a brief summary of known facts of CDC domains, including Hardy's inequality, and review a previous work of existence of globally Hölder continuous solutions. Additionally, we apply the result to homogenization problems of -periodic coefficients and present a convergence rate estimate of norms.

Paper Structure

This paper contains 6 sections, 7 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Capacity density condition
  3. Uniformly elliptic equations on CDC domains
  4. Abstract homogenization
  5. Periodic homogenization
  6. Conclusion

Key Result

Theorem 3.1

Let $\Omega$ be a bounded CDC domain. Assume that $A$ satisfis eqn:uniformly_elliptic. Then, for any $0 \le \mu \in \mathsf{M}^{\alpha}(\Omega)$, there exists $U \in H^{1}_{\mathrm{loc}}(\Omega) \cap C(\Omega)$ satisfying where $C$ and $\alpha_{0}$ are positive constants depending only on $n$, $L / \lambda$, $\alpha$ and $\gamma$.

Theorems & Definitions (19)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Theorem 3.1: hara2023global
  • Theorem 3.2: hara2023global
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • ...and 9 more