Solutions of the divergence and Korn inequalities on domains with an external cusp

TL;DR

This paper establishes the existence of divergence solutions and Korn inequalities in weighted Sobolev spaces for domains with external cusps, extending classical results to these singular geometries.

Contribution

It proves optimal weighted Sobolev space solutions for divergence and Korn inequalities on cusp domains, and demonstrates existence and uniqueness for Stokes equations in these settings.

Findings

Existence of divergence solutions in weighted Sobolev spaces for cusp domains

Optimal weighted Korn inequalities for domains with external cusps

Existence and uniqueness of Stokes solutions in cuspidal domains

Abstract

This paper deals with solutions of the divergence for domains with external cusps. It is known that the classic results in standard Sobolev spaces, which are basic in the variational analysis of the Stokes equations, are not valid for this class of domains. For some bounded domains $\Omega\subset{\mathbb R}^n$ presenting power type cusps of integer dimension $m\le n-2$, we prove the existence of solutions of the equation ${div}{\bf u}=f$ in weighted Sobolev spaces, where the weights are powers of the distance to the cusp. The results obtained are optimal in the sense that the powers cannot be improved. As an application, we prove existence and uniqueness of solutions of the Stokes equations in appropriate spaces for cuspidal domains. Also, we obtain weighted Korn type inequalities for this class of domains.