Article

Calculus of Variations and Partial Differential Equations

, Volume 34, Issue 3, pp 307-320

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

  • Fernando CharroAffiliated withDepartamento de Matemáticas, U. Autonoma de Madrid Email author 
  • , Jesus García AzoreroAffiliated withDepartamento de Matemáticas, U. Autonoma de Madrid
  • , Julio D. RossiAffiliated withDepartamento de Matemática, FCEyN, U. de Buenos Aires, Ciudad Universitaria

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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