Regularity for elliptic equations with monomial weights

TL;DR

This work develops a comprehensive regularity theory for elliptic equations with monomial weights $\omega^a(y)=\prod_{i=1}^n y_i^{a_i}$, degenerate or singular on orthogonal hyperplanes $\Sigma_i$. It introduces a robust weighted functional framework via regularized weights $\omega^a_{\varepsilon}$ and weighted Sobolev spaces, proves 2-admissibility to apply De Giorgi–Nash–Moser, and uses a regularization-approximation combined with blow-up and Liouville theorems to obtain $C^{0,\alpha}$ and $C^{1,\alpha}$ estimates up to corner intersections for conormal problems with variable coefficients. The paper also provides isotropic-homogeneous smoothness results and applies the theory to monomial CKN inequalities, highlighting the relevance to weighted isoperimetric and geometric analyses. Finally, higher-order regularity is established in special cases, including $C^{3,\alpha}$ regularity under additional structural assumptions and $C^{\infty}$ regularity for isotropic homogeneous problems, with implications for boundary value problems in domains with orthogonal corners.

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 $d\geq2$, the number of orthogonally crossing hyperplanes $1\leq n\leq d$ and the generic variable point $z=(x,y)\in\mathbb R^{d-n}\times\mathbb R^n$, then the weight is given by $ω(y)=\prod_{i=1}^ny_i^{a_i}$ with $a_i>-1$, $y_i=\mathrm{dist}(z,Σ_i)$ 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.

Theorems & Definitions (62)

• Theorem 1.2: Regularity for elliptic equations degenerating along orthogonal hyperplanes
• Theorem 2.1: Regularity in domains with corners
• Remark 2.2
• Theorem 2.3: Liouville Theorem in the orthant
• Remark 2.4
• Lemma 2.5: A priori estimates in domains with corners
• proof : Proof of Theorem \ref{['L:schuader:unifell']}
• Remark 3.1
• Lemma 3.2
• proof