2012/06/28 by S. Challal, Abdeslem Lyaghfouri, Challal, S. +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1206.6641
openalex publication_date 2012/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C1 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; 11.