A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman
In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div(g(ι∇u ει/ι∇uει)∇u ε) = βε(uε), u ε ≥ 0. A solution to (Pε) is a function uε Ε W1,G(Ω) ∩ L&infin(Ω) such that ∫ωg(ι∇uει) ∇u ε/ι∇uει ∇ℓ dx = ̄ ∫ω ℓβε(uε) dx for every ℓ Ε C&infin 0(Ω). H...
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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_00361410_v41_n1_p318_Sandra |
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| Sumario: | In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div(g(ι∇u ει/ι∇uει)∇u ε) = βε(uε), u ε ≥ 0. A solution to (Pε) is a function uε Ε W1,G(Ω) ∩ L&infin(Ω) such that ∫ωg(ι∇uει) ∇u ε/ι∇uει ∇ℓ dx = ̄ ∫ω ℓβε(uε) dx for every ℓ Ε C&infin 0(Ω). Here βε (s) = 1/εβ(s/ε), with β Ε Lip(rdbl), β > 0in(0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C 1, α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [ Comm. Partial Differential Equations, 16 (1991), pp. 311-361]. © 2009 Society for Industrial and Applied Mathematics. |
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