Analyticity up to the boundary for the divergence equation

TL;DR

This paper proves that solutions to the divergence equation in bounded analytic domains are analytic up to the boundary if the source term is analytic, extending regularity results to arbitrary bounded analytic domains.

Contribution

It establishes the analyticity up to the boundary for solutions of the divergence equation in arbitrary bounded analytic domains when the source term is analytic.

Findings

Solutions are analytic on the closure of the domain.

Analytic regularity holds up to the boundary.

Results apply to arbitrary bounded analytic domains.

Abstract

We address analytic regularity for the divergence equation $\text{div}\, u = f$ in $\Omega$, with $u=0$ on $\partial\Omega$, where $\Omega$ is an arbitrary bounded analytic domain and $\int_{\Omega} f\,dx=0$. If $f$ is analytic on the closure of $\Omega$, then we prove that there exists a solution that is analytic on the closure of $\Omega$.