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

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 ( − div(|y| aA(x, y)∇u) = |y| a f + div(|y| aF), u = ψ, on Σ0, where (x, y) ∈ R d−n × R n , 2 ≤ n ≤ d, a + n ∈ (0, 2), and Σ0 = {|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 Σ0, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a blow-up argument.