The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains

TL;DR

This paper establishes Schauder estimates for weighted elliptic equations with singular coefficients on lower dimensional manifolds, using perforated domain approximations and blow-up techniques.

Contribution

It introduces a novel approach combining perforated domain methods and Liouville theorems to derive regularity estimates for degenerate elliptic equations on lower dimensional sets.

Findings

Established $C^{0,eta}$ and $C^{1,eta}$ regularity up to the lower dimensional boundary.

Developed a new framework using perforated domain approximations for singular elliptic problems.

Proved Liouville-type theorems critical for regularity analysis.

Abstract

In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem \[\begin{cases} -{\rm div}(|y|^a A(x,y) \nabla u) = |y|^a f + {\rm div}(|y|^a F), \\ u = \psi, \quad \text{ on } \Sigma_0, \end{cases} \] where $(x,y) \in \mathbb{R}^{d-n} \times \mathbb{R}^n$, $2 \leq n \leq d$, $a + n \in (0,2)$, and $\Sigma_0 = \{|y| = 0\}$ is the lower dimensional manifold where the equation loses uniform ellipticity. Our primary objective is to establish $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity estimates up to $\Sigma_0$, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a fine blow-up argument.