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Browsing Glendon by Subject "Applied mathematics"
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Item Open Access On the regularity of the free boundary for quasilinear obstacle problems(European Mathematical Society - EMS - Publishing House, 2014-09-19) Challal, Samia; Lyaghfouri, Abdeslem; Rodrigues, José Francisco; Teymurazyan, RafayelWe 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, 1 < p < ∞; (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.Item Open Access Porosity of free boundaries in A-obstacle problems(Elsevier, 2009-04-08) Challal, S; Lyaghfouri, AWe establish the exact growth of the solution of the A-obstacle problem near the free boundary from which we deduce its porosity.Item Open Access Regularity results for a quasilinear free boundary problem(Vilnius Gediminas Technical University, 2020-05-13) Challal, Samia; Lyaghfouri, AbdeslemIn this paper we prove local interior and boundary Lipschitz continuity of the solutions of a quasilinear free boundary problem. We also show that the free boundary is the union of graphs of lower semi-continuous functions.