Low regularity results for degenerate Poisson problems

Abstract

In this paper we study the Poisson problem, \[ \begin{cases} -{\rm div}(d^β\nabla u)=f&{\rm in}\ Ω\\ u=0&{\rm on}\ \partialΩ, \end{cases} \] where $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$, $N\ge 2$ is a smooth bounded domain, $f$ is a continuous function, ${\hat{I}}^2<1$, and $d(x)=dist(x,\partial \hat{I}\text{c◯})$. We describe the behaviour of $u$ near $\partial \hat{I}\text{c◯}$ and discuss some of its regularity properties.

Updated version with some assumptions removed in Theorem 1.4. We thank Sergio Polidoro for the references that made this improvement possible