A game theoretical approach for a nonlinear system driven by elliptic operators
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ORIGINAL PAPER
A game theoretical approach for a nonlinear system driven by elliptic operators Alfredo Miranda1 • Julio D. Rossi1 Received: 9 March 2020 / Accepted: 6 May 2020 Ó Springer Nature Switzerland AG 2020
Abstract In this paper we find viscosity solutions to an elliptic system governed by two different operators (the Laplacian and the infinity Laplacian) using a probabilistic approach. We analyze a game that combines the tug-of-war with random walks in two different boards. We show that these value functions converge uniformly to a viscosity solution of the elliptic system as the step size goes to zero. In addition, we show uniqueness for the elliptic system using pure PDE techniques. Mathematics Subject Classification 35J94 35J47 35J60
1 Introduction Our goal in this paper is to describe a probabilistic game whose value functions approximate viscosity solutions to the following elliptic system: 8 1 > > x 2 X; D1 uðxÞ þ uðxÞ vðxÞ ¼ 0 > > 2 > > < j x 2 X; DvðxÞ þ vðxÞ uðxÞ ¼ 0 ð1Þ 2 > > > > uðxÞ ¼ f ðxÞ x 2 oX; > > : vðxÞ ¼ gðxÞ x 2 oX; here j [ 0 is a constant that can be chosen adjusting the parameters of the game. The domain X RN is assumed to be bounded and satisfy the uniform exterior ball property, that is, there is h [ 0 such that for all y 2 oX there exists a closed ball of radius h that only
This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira. & Julio D. Rossi [email protected] Alfredo Miranda [email protected] 1
Departamento de Matema´tica, FCEyN, Universidad de Buenos Aires, Pabellon I, Ciudad Universitaria (1428), Buenos Aires, Argentina SN Partial Differential Equations and Applications
14 Page 2 of 41
SN Partial Differ. Equ. Appl. (2020)1:14
touches X at y. This means that, for each y 2 oX there exists a zy 2 RN nX such that Bh ðzy Þ \ X ¼ fyg. The boundary data f and g are assumed to be Lipschitz functions. Notice that this system involves two differential operators, the usual Laplacian D/ ¼
N X
oxi xi /
i¼1
and the normalized infinity Laplacian (see [9]) N r/ r/ 1 X 2 ; ¼ oxi /oxi xj /oxj /; D1 / ¼ D / 2 jr/j jr/j jr/j i;j¼1 that is a 1-homogeneous, second order, degenerate elliptic operator. This system (1) is not variational (there is no associated energy). Therefore, to find solutions one possibility is to use monotonicity methods (Perron’s argument). Here we will look at the system in a different way and to obtain existence of solutions we find an approximation using game theory. This approach not only gives existence of solutions but it also provide us with a description that yield some light on the behaveiour of the solutions. At this point we observe that we will understand solutions to the system in the viscosity sense, this is natural since the infinity Laplacian is not variational (see Sect. 2 for the precise definition). The fundamental works by Doob, Feller, Hunt, Kakutani, Kolmogorov and many others show the deep connection between classical potential theory and probabil
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