Regularity for elliptic equations with monomial weights

TL;DR

This paper investigates the regularity of solutions to elliptic equations with monomial weights that are degenerate or singular along hyperplanes, establishing Hölder estimates and smoothness results, and applying these to inequalities.

Contribution

It introduces new regularity results for elliptic equations with monomial weights, including estimates up to corners and applications to inequalities, extending previous understanding of degenerate elliptic problems.

Findings

Established $C^{0,eta}$ and $C^{1,eta}$ estimates near hyperplane intersections.

Proved smoothness of solutions for isotropic, homogeneous cases.

Applied results to Caffarelli-Kohn-Nirenberg inequalities with monomial weights.

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 $\omega(y)=\prod_{i=1}^ny_i^{a_i}$ with $a_i>-1$, $y_i=\mathrm{dist}(z,\Sigma_i)$ and $\Sigma_i=\{y_i=0\}$. We prove $C^{0,\alpha}$ and $C^{1,\alpha}$ 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.