The obstacle problem for singular quasi-linear elliptic equations

TL;DR

This paper proves the existence and regularity of solutions to a singular obstacle p-Laplacian problem with discontinuous reactions, extending regularity results under certain conditions.

Contribution

It establishes existence and regularity of solutions for a singular obstacle p-Laplacian problem with discontinuous reactions, including boundary regularity under differentiability assumptions.

Findings

Solutions exist even with singular, discontinuous reactions.

Solutions are locally $C^{1,eta}$ away from the contact set.

Under obstacle differentiability, solutions are globally $C^{1,eta}$.

Abstract

Existence of solutions to an obstacle $p$-Laplacian problem exhibiting a singular, discontinuous reaction is proved. The reaction term may be discontinuous in a Lebesgue-negligible set. Moreover, solutions are shown to be locally $C^{1,\alpha}$ far away from the contact set. Under a differentiability hypothesis on the obstacle, solutions belong to $C^{1,\alpha}(\overline{\Omega})$.