Low regularity results for degenerate Poisson problems
Marta Calanchi, Massimo Grossi
TL;DR
We study the degenerate Poisson problem $L_beta(u) = -div(d^beta grad u) = f$ on a smooth bounded domain with $d(x)=dist(x,\partial\Omega)$ and $\beta<1$. The authors develop barrier-based, maximum-principle arguments in a tubular neighborhood of the boundary to derive explicit two-sided boundary bounds for $u$, revealing sharp boundary asymptotics of the form $u \sim d^{1-\beta}$ up to logarithmic corrections, and show an improved bound in convex domains. They place the problem in the weighted Sobolev framework and establish Sobolev regularity thresholds, notably that $u \in W^{1,2}_0(\Omega)$ if and only if $\beta<\frac{1}{2}$, while the Hölder regularity is limited by $\alpha \le \min\{1,1-\beta\}$ for $\beta>0$ and is shown to be sharp. The results highlight how boundary degeneracy, via the weight $d^\beta$, governs both boundary behavior and functional-analytic regularity, with implications for theory and numerical treatment of degenerate elliptic equations and potential extensions to broader boundary-behavior weights.
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 $Ω\subset\mathbb R^N$, $N\ge2$ is a smooth bounded domain, $f$ is a continuous function, $β< 1$, and $d(x)=dist(x,\partialΩ)$. We describe the behaviour of $u$ near $\partialΩ$ and discuss some of its regularity properties.