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# Interfaces and Free Boundaries

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**Volume 16, Issue 3, 2014, pp. 359–394**

**DOI: 10.4171/IFB/323**

Published online: 2014-09-19

On the regularity of the free boundary for quasilinear obstacle problems

Samia Challal^{[1]}, Abdeslem Lyaghfouri

^{[2]}, José Francisco Rodrigues

^{[3]}and Rafayel Teymurazyan

^{[4]}(1) York University, Toronto, Canada

(2) The Fields Institute for Research in Mathematical Sciences, Toronto, Canada

(3) FC Universidade de Lisboa, Portugal

(4) Universidade de Lisboa, Portugal

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 $p(x)>1$.

*Keywords: *Obstacle problem, regularity of the free boundary, quasi-linear elliptic operators, heterogeneous $p$-Laplacian

Challal Samia, Lyaghfouri Abdeslem, Rodrigues José Francisco, Teymurazyan Rafayel: On the regularity of the free boundary for quasilinear obstacle problems. *Interfaces Free Bound.* 16 (2014), 359-394. doi: 10.4171/IFB/323