A free boundary problem for the p (x)-Laplacian
We consider the optimization problem of minimizing ∫ Ω frac(1, p (x)) | ∇ u | p (x) + λ (x) χ {u > 0} d x in the class of functions W 1, p ({dot operator}) (Ω) with u - φ 0 ∈ W 0 1, p ({dot operator}) (Ω), for a given φ 0 ≥ 0 and bounded. W 1, p ({dot operator}) (Ω) is the class of weakly dif...
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| Autores principales: | , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_0362546X_v72_n2_p1078_Bonder |
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| Sumario: | We consider the optimization problem of minimizing ∫ Ω frac(1, p (x)) | ∇ u | p (x) + λ (x) χ {u > 0} d x in the class of functions W 1, p ({dot operator}) (Ω) with u - φ 0 ∈ W 0 1, p ({dot operator}) (Ω), for a given φ 0 ≥ 0 and bounded. W 1, p ({dot operator}) (Ω) is the class of weakly differentiable functions with ∫ Ω | ∇ u | p (x) d x < ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω ∩ ∂ {u > 0}, is a regular surface. © 2009 Elsevier Ltd. All rights reserved. |
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