Boundary C^{\alpha}-regularity for solutions of elliptic equations with distributional coefficients

TL;DR

This paper establishes boundary regularity for solutions of elliptic equations with distributional coefficients, extending interior regularity results to boundary points under certain measure and decay conditions.

Contribution

It proves boundary pointwise $C^{0}$-regularity for elliptic equations with distributional coefficients, generalizing previous interior regularity results.

Findings

Solutions are continuous at boundary points under measure and decay conditions.

Boundary regularity is achieved for elliptic equations with distributional coefficients.

The results extend classical regularity theory to more general coefficients.

Abstract

In this paper, we prove the boundary pointwise $C^{0}$-regularity of weak solutions for Dirichlet problem of elliptic equations in divergence form with distributional coefficients, where the boundary value equals to zero. This is a generalization of the interior case. If $\Omega$ satisfies some measure condition at one boundary point, the bilinear mapping $\langle V\cdot,\cdot\rangle$ generalized by distributional coefficient $V$ can be controlled by a constant sufficiently small, the nonhomogeneous terms satisfy some Dini decay conditions, then the solution is continuous at this point in the $L^{2}$ sense.