Differential operators on domains with conical points: precise uniform regularity estimates

TL;DR

This paper establishes uniform regularity estimates for strongly elliptic differential operators with singular coefficients on domains with conical points, extending classical results to more general singular coefficient settings.

Contribution

It provides uniform estimates and regularity results for elliptic operators with singular coefficients on conical domains, broadening classical well-posedness theory.

Findings

Uniform estimates on inverse operators

Regularity results in weighted Sobolev spaces

Extension of classical well-posedness to singular coefficients

Abstract

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated Poisson problem with mixed boundary conditions. The coefficients and the solutions belong to (suitable) weighted Sobolev spaces. The space of coefficients is a Banach space that contains, in particular, the space of smooth functions. Hence, our results extend classical well-posedness results for strongly elliptic equations in domains with conical points to problems with singular coefficients. We furthermore provide precise uniform estimates on the norms of the solution operators.