Tug-of-wargames and the infinity Laplacian with spatial dependence

Ivana Gómez; Julio D. Rossi

doi:10.3934/cpaa.2013.12.1959

Article Contents

2013, Volume 12, Issue 5: 1959-1983. Doi: 10.3934/cpaa.2013.12.1959

This issue Previous Article Elliptic equations with cylindrical potential and multiple critical exponents Next Article Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential

Tug-of-war games and the infinity Laplacian with spatial dependence

Ivana Gómez^1, and
Julio D. Rossi^2,

1.
Instituto de Matemática Aplicada del Litoral (IMAL), CONICET-UNL, Departamento de Matemática, Facultad de Ingeniería Química, UNL, Güemes 3450, S3000GLN Santa Fe
2.
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, 1428 – Buenos Aires

Received: December 31, 2011

Revised: September 30, 2012

Published: August 2013

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Abstract

In this paper we look for PDEs that arise as limits of values of tug-of-war games when the possible movements of the game are taken in a family of sets that are not necessarily Euclidean balls. In this way we find existence of viscosity solutions to the Dirichlet problem for an equation of the form $- \langle D^2 v\cdot J_x(D v) ; J_x(Dv)\rangle (x) =0$, that is, an infinity Laplacian with spatial dependence. Here $J_x (Dv(x))$ is a vector that depends on the spatial location and the gradient of the solution.

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Mathematics Subject Classification: Primary: 35B50, 35J25, 35J70, 49N70, 91A15, 91A24.

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