A right inverse of the divergence for planar H\"older-$\alpha$ domains

TL;DR

This paper constructs right inverses of the divergence operator for planar H"older-$eta$ domains, enabling solutions to the Stokes equations with weaker boundary conditions in weighted Sobolev spaces, especially for domains with cusps.

Contribution

It proves the existence of divergence right inverses in weighted spaces for planar H"older-$eta$ domains, extending classical results to non-Lipschitz domains with cusps.

Findings

Existence of divergence right inverses in weighted spaces for planar H"older-$eta$ domains.

Weak boundary conditions are shown to be equivalent to standard conditions in cusp domains.

Solutions to the Stokes equations are obtained in weighted Sobolev spaces with some $L^r$ integrability, $r<2$.

Abstract

If $\Omega\subset\R^n$ is a bounded domain, the existence of solutions ${\bf u}\in H^1_0(\Omega)^n$ of ${div} {\bf u} = f$ for $f\in L^2(\Omega)$ with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution $({\bf u},p)\in H^1_0(\Omega)^n\times L^2(\Omega)$, where ${\bf u}$ is the velocity and $p$ the pressure. It is known that the above mentioned result holds when $\Omega$ is a Lipschitz domain and that it is not valid for arbitrary H\"older-$\alpha$ domains. In this paper we prove that if $\Omega$ is a planar simply connected H\"older-$\alpha$ domain, there exist right inverses of the divergence which are continuous in appropriate weighted spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. In our results, the zero boundary condition is replaced by a weaker one. For the particular case of domains with an external cusp of power type, we prove that our weaker boundary condition is equivalent to the standard one. In this case we show the well posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution $({\bf u},p)\in H^1_0(\Omega)^n\times L^r(\Omega)$ for some $r<2$ depending on the power of the cusp.