A mixed problem for the infinity Laplacian via Tug-of-War games

  • Fernando Charro
  • Jesus García Azorero
  • Julio D. Rossi
Article

DOI: 10.1007/s00526-008-0185-2

Cite this article as:
Charro, F., García Azorero, J. & Rossi, J.D. Calc. Var. (2009) 34: 307. doi:10.1007/s00526-008-0185-2

Abstract

In this paper we prove that a function \({ u\in\mathcal{C}(\overline{\Omega})}\) is the continuous value of the Tug-of-War game described in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear) if and only if it is the unique viscosity solution to the infinity Laplacian with mixed boundary conditions
$$\left\{ \begin{aligned}-\Delta_{\infty}u(x)=0 \quad & {\rm in} \, \Omega,\\ \frac{\partial u}{\partial n}(x)=0 \quad \quad & {\rm on} \, \Gamma_N,\\ u(x)=F(x) \quad & {\rm on}\, \Gamma_D. \end{aligned} \right.$$
By using the results in Y. Peres et al. (J. Am. Math. Soc., 2008, to appear), it follows that this viscous PDE problem has a unique solution, which is the unique absolutely minimizing Lipschitz extension to the whole \({\overline{\Omega}}\) (in the sense of Aronsson (Ark. Mat. 6:551–561, 1967) and Y. Peres et al. (J. Am. Math. Soc., 2008, to appear)) of the Lipschitz boundary data \({F:\Gamma_D \to \mathbb R }\).

Mathematics Subject Classification (2000)

35J60 91A05 49L25 35J25 

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Fernando Charro
    • 1
  • Jesus García Azorero
    • 1
  • Julio D. Rossi
    • 2
  1. 1.Departamento de MatemáticasU. Autonoma de MadridMadridSpain
  2. 2.Departamento de MatemáticaFCEyN, U. de Buenos Aires, Ciudad UniversitariaBuenos AiresArgentina

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