On the regularity of the free boundary for quasilinear obstacle problems (Q467368)

Summary: We extend basic regularity of the free boundary of the obstacle problem to some classes of heterogeneous quasilinear elliptic operators with variable growth that include, 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 the data we show, in two different cases, that up to a negligible singular set of null perimeter the free boundary is the union of an at most 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 that 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 \(p(x)>1\).