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

TL;DR

The paper studies local weak solutions to the wildly degenerate elliptic equation $\mathrm{div}\bigl(a(x)(|Du|-1)_+^{p-1}\frac{Du}{|Du|}\bigr)=b(x,u)$ with $p\ge2$ and explicit $u$-dependence in the right-hand side. The authors develop a regularization/frozen-data framework, establish higher differentiability for the function $H_{p/2}(Du)$ by deriving robust a priori estimates, and then pass to the limit via mollification to obtain the same regularity for the original problem. The key contributions include minimal regularity assumptions on $a(x)$ ($a\in W^{1,n}_{\text{loc}}$ with $m<a<M$ a.e.) and structured conditions on $b(x,u)$ that accommodate explicit $u$-dependence, yielding $H_{p/2}(Du)\in W^{1,2}_{\text{loc}}(\Omega)$. This extends existing results for widely degenerate problems to settings where the source term depends on $u$, with implications for models arising in optimal transport with congestion and related nonlinear PDEs.

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 }Ω, \end{equation} where $2\leq p<\infty,\textbf{ } Ω$ 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 $u.$

Theorems & Definitions (19)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Definition 2.7
• Proposition 2.8
• Lemma 2.9