Lipschitz continuity results for a class of obstacle problems

TL;DR

This paper establishes Lipschitz continuity for solutions to a class of obstacle problems with p-growth conditions, using a novel linearization and Moser iteration approach, advancing regularity theory for nonlinear elliptic equations.

Contribution

It introduces a new linearization technique combined with Moser iteration to prove Lipschitz regularity for obstacle problems with standard p-growth, a first in non-autonomous functionals.

Findings

Lipschitz continuity of solutions established

New linearization method applied to obstacle problems

Simplified identification of Radon measure in linearization

Abstract

We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of $p$-type, $p \geq 2$. The main novelty is the use of a linearization technique going back to [28] in order to interpret our constrained minimizer as a solution to a nonlinear elliptic equation, with a bounded right-hand side. This leads us to start a Moser iteration scheme which provides the $L^\infty$ bound for the gradient. The application of a recent higher differentiability result [24] allows us to simplify the procedure of the identification of the Radon measure in the linearization technique employed in [32]. To our knowledge, this is the first result for non-autonomous functionals with standard growth conditions in the direction of the Lipschitz regularity.