A $W^n_2$-Theory of Elliptic and Parabolic Partial Differential Systems in $C^1$ domains

TL;DR

This paper develops a weighted Sobolev space framework for second-order elliptic and parabolic PDE systems in $C^1$ domains, allowing solutions with boundary blow-up and coefficients with significant oscillation or blow-up.

Contribution

It introduces a $W^n_2$-theory that handles boundary blow-up solutions and highly oscillatory coefficients in $C^1$ domains for elliptic and parabolic systems.

Findings

Existence and uniqueness of solutions in weighted Sobolev spaces.

Framework accommodates solutions with derivatives blowing up near the boundary.

Handles coefficients that oscillate or blow up near the boundary.

Abstract

In this paper second-order elliptic and parabolic partial differential systems are considered on $C^1$ domains. Existence and uniqueness results are obtained in terms of Sobolev spaces with weights so that we allow the derivatives of the solutions to blow up near the boundary. The coefficients of the systems are allowed to substantially oscillate or blow up near the boundary.