# On the regularity of the free boundary for quasilinear obstacle problems

### Samia Challal

York University, Toronto, Canada### Abdeslem Lyaghfouri

The Fields Institute for Research in Mathematical Sciences, Toronto, Canada### José Francisco Rodrigues

FC Universidade de Lisboa, Portugal### Rafayel Teymurazyan

Universidade de Lisboa, Portugal

## 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 $p(x)>1$.

## Cite this article

Samia Challal, Abdeslem Lyaghfouri, José Francisco Rodrigues, Rafayel Teymurazyan, On the regularity of the free boundary for quasilinear obstacle problems. Interfaces Free Bound. 16 (2014), no. 3, pp. 359–394

DOI 10.4171/IFB/323