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

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 $(x,y)\in {\mathbb{R}}^{d-n}\times {\mathbb{R}}^n$, $2\le n\le d$, $a+n\in (0,2)$, 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.

43 pages

References in corpus (5)

• Higher order boundary Harnack principle via degenerate equations
• Schauder estimates for parabolic equations with degenerate or singular weights
• Schauder type estimates for degenerate or singular elliptic equations with DMO coefficients
• Higher order Schauder estimates for degenerate or singular parabolic equations
• Schauder estimates for elliptic equations degenerating on lower dimensional manifolds