A limiting free boundary problem with gradient constraint and Tug-of-War games
In this manuscript we deal with regularity issues and the asymptotic behaviour (as p→ ∞) of solutions for elliptic free boundary problems of p- Laplacian type (2 ≤ p< ∞): -Δpu(x)+λ0(x)χ{u>0}(x)=0inΩ⊂RN,with a prescribed Dirichlet boundary data, where λ> 0 is a bounded function and Ω is a re...
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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_03733114_v_n_p_Blanc |
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| Sumario: | In this manuscript we deal with regularity issues and the asymptotic behaviour (as p→ ∞) of solutions for elliptic free boundary problems of p- Laplacian type (2 ≤ p< ∞): -Δpu(x)+λ0(x)χ{u>0}(x)=0inΩ⊂RN,with a prescribed Dirichlet boundary data, where λ> 0 is a bounded function and Ω is a regular domain. First, we prove the convergence as p→ ∞ of any family of solutions (up)p≥2, as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, {max{-Δ∞u∞,-|∇u∞|+χ{u∞>0}}=0inΩ∩{u∞≥0}u∞=Fon∂Ω.Next, we obtain uniqueness for solutions to this limit problem. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game. © 2019, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature. |
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