Low regularity results for degenerate Poisson problems

In this paper we study the Poisson problem,$\{\begin{array}{cc}-\mathrm{div}\left(d^{\beta }\nabla u\right)=f& \mathrm{in}\Omega \\ u=0& \mathrm{on}\partial \Omega ,\end{array}$ where $\Omega \subset R^N$, $N\ge 2$ is a smooth bounded domain, f is a continuous function, $\beta <1$, and $d\left(x\right)=dist(x,\partial \Omega )$. We describe the behaviour of u near ∂Ω and discuss some of its regularity properties.