Regularity for a strongly degenerate equation with explicit $u$-dependence

TL;DR

This paper proves higher differentiability of solutions to a class of degenerate elliptic PDEs with explicit dependence on the solution, extending previous results by including the right-hand side's dependence on u.

Contribution

It introduces a higher differentiability result for degenerate elliptic equations with explicit u-dependence on the right-hand side, a novel aspect compared to prior work.

Findings

Established higher differentiability for the gradient composition under new conditions.

Extended regularity results to equations with solution-dependent right-hand sides.

Provided assumptions on data and coefficients for the regularity to hold.

Abstract

We consider local weak solutions of widely degenerate elliptic PDEs of the type \begin{equation} \label{equazione mia} \mathrm{div}\Biggl(a(x)(|Du|-1)^{p-1}_+\frac{Du}{|Du|}\Biggr)=b(x,u) \ \ \text{ in }\Omega, \end{equation} where $2\leq p<\infty,\textbf{ } \Omega$ is an open subset of $\mathbb{R}^n,n>2,$ and $( \ \cdot \ )_+$ stands for the positive part. We establish a higher differentiability result for the composition of the gradient with a suitable function that vanishes in the unit ball for the gradient, under suitable assumptions on the datum $b(x,u)$ and the coefficient $a(x).$ The novelty here with respect to previous papers on the subject is that the right hand side explicitly depends on the solution $u.$