A game theoretical approach for a nonlinear system driven by elliptic operators. (English) Zbl 1455.35081
Summary: 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.
MSC:
| 35J47 | Second-order elliptic systems |
| 35J94 | Elliptic equations with infinity-Laplacian |
| 35J91 | Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian |
| 35D40 | Viscosity solutions to PDEs |
Keywords:
elliptic system; Laplacian; infinity Laplacian; viscosity solutions; probabilistic approachReferences:
| [1] | Antunovic, T.; Peres, Y.; Sheffield, S.; Somersille, S., Tug-of-war and infinity Laplace equation with vanishing Neumann boundary condition, Commun. Partial Differ. Equ., 37, 10, 1839-1869 (2012) · Zbl 1268.35065 |
| [2] | Armstrong, SN; Smart, CK, An easy proof of Jensen’s theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differ. Equ., 37, 3-4, 381-384 (2010) · Zbl 1187.35104 |
| [3] | Barles, G.; Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term, Commun. Partial Differ. Equ., 26, 11-12, 2323-2337 (2001) · Zbl 0997.35023 |
| [4] | Birindelli, I.; Galise, G.; Ishii, H., A family of degenerate elliptic operators: maximum principle and its consequences, Ann. Inst. H. Poincare Anal. Non Lineaire, 35, 2, 417-441 (2018) · Zbl 1390.35079 |
| [5] | Blanc, P.; Rossi, JD, Games for eigenvalues of the Hessian and concave/convex envelopes, J. Math. Pures Appl., 127, 192-215 (2019) · Zbl 1423.35118 |
| [6] | Blanc, P., Rossi, J. D.: Game Theory and Partial Differential Equations. De Gruyter Series in Nonlinear Analysis and Applications Vol. 31 (2019). ISBN 978-3-11-061925-6. ISBN 978-3-11-062179-2 (eBook) · Zbl 1430.91001 |
| [7] | Blanc, P.; Manfredi, JJ; Rossi, JD, Games for Pucci’s maximal operators, J. Dyn. Games, 6, 4, 277-289 (2019) · Zbl 1439.35104 |
| [8] | Charro, F.; Garcia Azorero, J.; Rossi, JD, A mixed problem for the infinity laplacian via Tug-of-War games, Calc. Var. Partial Differ. Equ., 34, 3, 307-320 (2009) · Zbl 1173.35459 |
| [9] | Crandall, MG, A Visit with the \(\infty \)-Laplace Equation, Calculus of variations and Nonlinear Partial Differential Equations. Lecture Notes in Mathematics, 75-122 (2008), Berlin: Springer, Berlin · Zbl 1357.49112 |
| [10] | Crandall, MG; Ishii, H.; Lions, PL, User’s guide to viscosity solutions of second order partial differential equations, Bull. Am. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015 |
| [11] | Doob, JL, What is a martingale?, Am. Math. Mon., 78, 5, 451-463 (1971) · Zbl 0215.25801 |
| [12] | Doob, JL, Classical Potential Theory and Its Probabilistic Counterpart. Classics in Mathematics (2001), Berlin: Springer, Berlin · Zbl 0990.31001 |
| [13] | Doob, JL, Semimartingales and subharmonic functions, Trans. Am. Math. Sot., 77, 86-121 (1954) · Zbl 0059.12205 |
| [14] | Hunt, GA, Markoff processes and potentials I, II, III, Ill. J. Math., 1, 44-93, 316-369 (1957) · Zbl 0100.13804 |
| [15] | Hunt, GA, Markoff processes and potentials I, II, III, Ill. J. Math., 2, 151-213 (1958) |
| [16] | Ishiwata, M., Magnanini, R., Wadade, H.: A natural approach to the asymptotic mean value property for the p-Laplacian. Calc. Var. Partial Differ. Equ. 56(4), Art. 97, 22 (2017) · Zbl 1378.35147 |
| [17] | Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Ration. Mech. Anal., 123, 1, 51-74 (1993) · Zbl 0789.35008 |
| [18] | Kac, M., Random walk and the theory of Brownian motion, Am. Math. Mon., 54, 7, 369-391 (1947) · Zbl 0031.22604 |
| [19] | Kakutani, S., Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad. Tokyo, 20, 706-714 (1944) · Zbl 0063.03107 |
| [20] | Kawohl, B.; Manfredi, JJ; Parviainen, M., Solutions of nonlinear PDEs in the sense of averages, J. Math. Pures Appl., 97, 3, 173-188 (2012) · Zbl 1236.35083 |
| [21] | Knapp, AW, Connection between Brownian motion and potential theory, J. Math. Anal. Appl., 12, 328-349 (1965) · Zbl 0138.40801 |
| [22] | Lindqvist, P.; Manfredi, JJ, On the mean value property for the \(p-\) Laplace equation in the plane, Proc. Am. Math. Soc., 144, 1, 143-149 (2016) · Zbl 1327.35124 |
| [23] | Luiro, H.; Parviainen, M.; Saksman, E., Harnack’s inequality for p-harmonic functions via stochastic games, Comm. Partial Differ. Equ., 38, 11, 1985-2003 (2013) · Zbl 1287.35044 |
| [24] | Manfredi, JJ; Parviainen, M.; Rossi, JD, An asymptotic mean value characterization for p-harmonic functions, Proc. Am. Math. Soc., 138, 3, 881-889 (2010) · Zbl 1187.35115 |
| [25] | Manfredi, JJ; Parviainen, M.; Rossi, JD, Dynamic programming principle for tug-of-war games with noise, ESAIM Control Opt. Calc. Var., 18, 81-90 (2012) · Zbl 1233.91042 |
| [26] | Manfredi, JJ; Parviainen, M.; Rossi, JD, On the definition and properties of p-harmonious functions, Ann. Scuola Nor. Sup. Pisa, 11, 215-241 (2012) · Zbl 1252.91014 |
| [27] | Mitake, H.; Tran, HV, Weakly coupled systems of the infinity Laplace equations, Trans. Am. Math. Soc., 369, 1773-1795 (2017) · Zbl 1359.35054 |
| [28] | Oksendal, B., Stochastic Differential Equations: An Introduction with Applications (2003), Berlin: Springer, Berlin · Zbl 1025.60026 |
| [29] | Peres, Y.; Schramm, O.; Sheffield, S.; Wilson, D., Tug-of-war and the infinity Laplacian, J. Am. Math. Soc., 22, 167-210 (2009) · Zbl 1206.91002 |
| [30] | Peres, Y.; Sheffield, S., Tug-of-war with noise: a game theoretic view of the \(p\)-Laplacian, Duke Math. J., 145, 1, 91-120 (2008) · Zbl 1206.35112 |
| [31] | Rossi, JD, Tug-of-war games and PDEs, Proc. R. Soc. Edim., 141A, 319-369 (2011) · Zbl 1242.35091 |
| [32] | Williams, D., Probability with Martingales (1991), Cambridge: Cambridge University Press, Cambridge · Zbl 0722.60001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
