On the Regularity of the Free Boundary for Quasilinear Obstacle Problems
| Autor: | Challal, S., Lyaghfouri, A., Rodrigues, J. F., Teymurazyan, R. |
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| Rok vydání: | 2012 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 Comment: 40 pages |
| Databáze: | arXiv |
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