Maximal operators for the P-laplacian family

We prove existence and uniqueness of viscosity solutions for the problem: max-Δp1u(x), -Δp2u(x) = f(x) in a bounded smooth domain Ω⊂ℝN with u=g on ∂Ω. Here -Δpu=(N+ p)-1|Du|2-pdiv (|Du|p-2Du) is the 1-homogeneous p-Laplacian and we assume that 2 ≤ p1; p2 ≤ ∞. This equation appears naturally when one...

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Detalles Bibliográficos
Autores principales:	Blanc, P., Pinasco, J.P., Rossi, J.D.
Formato:	JOUR
Materias:	Dirichlet boundary conditions Dynamic programming principle P-Laplacian Tug-of-war games
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_00308730_v287_n2_p257_Blanc
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

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spelling	todo:paper_00308730_v287_n2_p257_Blanc2023-10-03T14:40:47Z Maximal operators for the P-laplacian family Blanc, P. Pinasco, J.P. Rossi, J.D. Dirichlet boundary conditions Dynamic programming principle P-Laplacian Tug-of-war games We prove existence and uniqueness of viscosity solutions for the problem: max-Δp1u(x), -Δp2u(x) = f(x) in a bounded smooth domain Ω⊂ℝN with u=g on ∂Ω. Here -Δpu=(N+ p)-1\|Du\|2-pdiv (\|Du\|p-2Du) is the 1-homogeneous p-Laplacian and we assume that 2 ≤ p1; p2 ≤ ∞. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff ) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-ofwar game (without noise) or playing at random. Moreover, the operator max-Δp1u(x), -Δp2u(x) provides a natural analogue with respect to p- Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory. © 2017 Mathematical Sciences Publishers. Fil:Pinasco, J.P. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00308730_v287_n2_p257_Blanc
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Dirichlet boundary conditions Dynamic programming principle P-Laplacian Tug-of-war games
spellingShingle	Dirichlet boundary conditions Dynamic programming principle P-Laplacian Tug-of-war games Blanc, P. Pinasco, J.P. Rossi, J.D. Maximal operators for the P-laplacian family
topic_facet	Dirichlet boundary conditions Dynamic programming principle P-Laplacian Tug-of-war games
description	We prove existence and uniqueness of viscosity solutions for the problem: max-Δp1u(x), -Δp2u(x) = f(x) in a bounded smooth domain Ω⊂ℝN with u=g on ∂Ω. Here -Δpu=(N+ p)-1\|Du\|2-pdiv (\|Du\|p-2Du) is the 1-homogeneous p-Laplacian and we assume that 2 ≤ p1; p2 ≤ ∞. This equation appears naturally when one considers a tug-of-war game in which one of the players (the one who seeks to maximize the payoff ) can choose at every step which are the parameters of the game that regulate the probability of playing a usual tug-ofwar game (without noise) or playing at random. Moreover, the operator max-Δp1u(x), -Δp2u(x) provides a natural analogue with respect to p- Laplacians to the Pucci maximal operator for uniformly elliptic operators. We provide two different proofs of existence and uniqueness for this problem. The first one is based in pure PDE methods (in the framework of viscosity solutions) while the second one is more connected to probability and uses game theory. © 2017 Mathematical Sciences Publishers.
format	JOUR
author	Blanc, P. Pinasco, J.P. Rossi, J.D.
author_facet	Blanc, P. Pinasco, J.P. Rossi, J.D.
author_sort	Blanc, P.
title	Maximal operators for the P-laplacian family
title_short	Maximal operators for the P-laplacian family
title_full	Maximal operators for the P-laplacian family
title_fullStr	Maximal operators for the P-laplacian family
title_full_unstemmed	Maximal operators for the P-laplacian family
title_sort	maximal operators for the p-laplacian family
url	http://hdl.handle.net/20.500.12110/paper_00308730_v287_n2_p257_Blanc
work_keys_str_mv	AT blancp maximaloperatorsfortheplaplacianfamily AT pinascojp maximaloperatorsfortheplaplacianfamily AT rossijd maximaloperatorsfortheplaplacianfamily
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Maximal operators for the P-laplacian family

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