On the Regularity of the Free Boundary for Quasilinear Obstacle Problems

TL;DR

This paper investigates the regularity of the free boundary in obstacle problems involving heterogeneous quasilinear elliptic operators, including the $p(x)$-Laplacian, establishing conditions under which the free boundary is smooth or of measure zero.

Contribution

It extends regularity results to variable growth operators like the $p(x)$-Laplacian, showing free boundary smoothness and measure properties under new assumptions.

Findings

Free boundary is porous and of measure zero for $p(x)$-Laplacian type problems.

Under certain conditions, the free boundary is a union of countably many $C^1$ hypersurfaces.

The characteristic function of the coincidence set has bounded variation in specific cases.

Abstract

We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that includes, in particular, the $p(x)$-Laplacian. Under the assumption of Lipschitz continuity of the order of the power growth $p(x)>1$, we use the growth rate of the solution near the free boundary to obtain its porosity, which implies that the free boundary is of Lebesgue measure zero for $p(x)$-Laplacian type heterogeneous obstacle problems. Under additional assumptions on the operator heterogeneities and on data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of at most a countable family of $C^1$ hypersurfaces: i) by extending directly the finiteness of the $(n-1)$-dimensional Hausdorff measure of the free boundary to the case of heterogeneous $p$-Laplacian type operators with constant $p$; $1<p<\infty$; ii) by proving the characteristic function of the coincidence set is of bounded variation in the case of non degenerate or non singular operators with variable power growth $p(x)>1$.