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

TL;DR

The paper addresses the Dirichlet problem for a weighted elliptic operator with weight $|y|^a$ that degenerates on a lower-dimensional boundary $\Sigma_0$ when $a+n\in(0,2)$. It innovates by employing perforated-domain approximations, Liouville-type theorems, and blow-up analysis to obtain local $C^{0,\alpha}$ and $C^{1,\alpha}$ regularity up to $\Sigma_0$, and extends the framework to curved manifolds with distance-type weights $\delta^a$. The main results provide $\varepsilon$-uniform Hölder and Schauder estimates and reveal that, under suitable data assumptions, weak solutions exhibit sharp regularity up to the singular set, including conormal boundary conditions in the limit. This work connects non-uniform ellipticity, edge/stratified operator theory, and thin-boundary problems and offers a robust method for thin-domain regularity with potential applications to obstacle-type free boundary problems.

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 \[\begin{cases} -{\rm div}(|y|^a A(x,y) \nabla u) = |y|^a f + {\rm div}(|y|^a F), \\ u = ψ, \quad \text{ on } Σ_0, \end{cases} \] where $(x,y) \in \mathbb{R}^{d-n} \times \mathbb{R}^n$, $2 \leq n \leq d$, $a + n \in (0,2)$, and $Σ_0 = \{|y| = 0\}$ is the lower dimensional manifold where the equation loses uniform ellipticity. Our primary objective is to establish $C^{0,α}$ and $C^{1,α}$ regularity estimates up to $Σ_0$, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a fine blow-up argument.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Definition 2.5