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

2024/12/15 by Gabriele Fioravanti, Fioravanti, Gabriele · 1 citation
Computer Science · Mathematics · #35B44 #35B53 (secondary) #35B65 (primary) #35J25 #35J75 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2412.11294

openalex publication_date 2024/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

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 \begincases -\rm div(|y|^a A(x,y) ∇ u) = |y|^a f + \rm div(|y|^a F),
u = ψ, on Σ₀, \endcases where (x,y) ∈ ℝ^d-n × ℝⁿ, 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 fine blow-up argument.

Cited by

• On some divergence-form singular elliptic equations with codimension-two boundary: L^p-estimates

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