On the second-order regularity of solutions to widely singular or degenerate elliptic equations

Abstract

We consider local weak solutions to PDEs of the type \[ -\,\mathrm{div}\left((\vert Du\vert-λ)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\,\,\,\,\,\,\,\text{in}\,\,Ω, \] where $1<p<\infty$, $\Omega$ is an open subset of ${\mathbb{R}}^n$ for $n\ge 2$, $\lambda$ is a positive constant and $(\cdot {)}_{+}$ stands for the positive part. Equations of this form are widely degenerate for $p\ge 2$ and widely singular for $1<p<2$. We establish higher differentiability results for a suitable nonlinear function of the gradient $Du$ of the local weak solutions, assuming that $f$ belongs to the local Besov space $B_{p^{\prime },1,loc}^{(p-2)/p}(\Omega )$ when $p>2$, and that $f\in L_{loc}^{\frac{np}{n(p-1)+2-p}}(\Omega )$ if $1<p\le 2$. The conditions on the datum $f$ are essentially sharp. As a consequence, we obtain the local higher integrability of $Du$ under the same minimal assumptions on $f$. For $\lambda =0$, our results give back those contained in [12,28].

Annali di Matematica (2025)

References in corpus (2)

• Regularity results for a class of widely degenerate parabolic equations
• Besov regularity for a class of singular or degenerate elliptic equations