A weighted Sobolev regularity theory of the parabolic equations with measurable coefficients on conic domains in $R^d$

TL;DR

This paper develops a weighted Sobolev regularity theory for second-order parabolic equations with measurable coefficients on conic domains in , establishing existence, uniqueness, and high-order regularity results using mixed weights related to the domain geometry.

Contribution

It introduces a novel weighted Sobolev framework for parabolic equations on conic domains, providing sharp admissible ranges for weights and high-order regularity results.

Findings

Established existence and uniqueness of solutions.

Achieved arbitrary order Sobolev regularity.

Identified sharp ranges for weight parameters.

Abstract

We establish existence, uniqueness, and arbitrary order Sobolev regularity results for the second order parabolic equations with measurable coefficients defined on the conic domains $D$ of the type $$ D(M):=\left\{x\in R^d :\,\frac{x}{|x|}\in M\right\}, \quad \quad M \subset S^{d-1}. $$ We obtain the regularity results by using a system of mixed weights consisting of appropriate powers of the distance to the vertex and of the distance to the boundary. We also provide the sharp ranges of admissible powers of the distance to the vertex and to the boundary.