Regularity for elliptic equations with monomial weights

2025/11/20 by Gabriele Cora, Cora, Gabriele, Gabriele Fioravanti +5
Mathematics · #35B40 #35B44 #35B45 #35B53 #35B65 #35J70 #35J75 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2511.16516

openalex publication_date 2025/11/20 · openalex created_date 2025/11/23 · openalex updated_date 2026/07/28

Abstract

We study regularity properties for solutions to elliptic equations that are degenerate or singular along orthogonal hyperplanes. The degenerate ellipticity is carried out by a weight term which is the monomial product of different powers of the distance functions to each hyperplane; that is, given the space dimension d≥2, the number of orthogonally crossing hyperplanes 1≤ n≤ d and the generic variable point z=(x,y)∈\mathbb R^d-n×\mathbb Rⁿ, then the weight is given by ω(y)=∏_i=1ⁿy_i^a_i with a_i>-1, y_i=dist(z,Σ_i) and Σ_i=\y_i=0\. We prove C^0,α and C^1,α estimates up to the corners formed by the intersections of two or more hyperplanes, for solutions of the conormal problem with variable coefficients. This is done by a regularization-approximation procedure, a blow-up argument and Liouville theorems. Finally, we provide smoothness of solutions when the equation is isotropic and homogeneous, and we show an application to Caffarelli-Kohn-Nirenberg inequalities with monomial weights.

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