On elliptic equations in a half space or in convex wedges with irregular coefficients

TL;DR

This paper establishes $W^2_p$-estimates and solvability for second-order elliptic equations in half spaces and convex wedges with irregular coefficients, extending classical results to more general coefficient conditions.

Contribution

It provides new $W^2_p$-estimates and solvability results for elliptic equations with irregular coefficients in half spaces and convex domains, including cases with coefficients measurable in tangential directions.

Findings

Proved $W^2_p$-estimates for Dirichlet problems when p in (1,2].

Established solvability for Neumann problems when p in [2,∞).

Extended results to equations with coefficients having small mean oscillations.

Abstract

We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann problem when $p\in [2,\infty)$. We then extend these results to equations with more general coefficients, which are measurable in a tangential direction and have small mean oscillations in the other directions. As an application, we obtain the $W^2_p$-solvability of elliptic equations in convex wedge domains or in convex polygonal domains with discontinuous coefficients.