Nonlinear elliptic systems in stochastic game theory.

*(English)*Zbl 0531.93052This article considers systems of nonlinear partial differential equations arising from stochastic game theory in the sense of Nash. The main purpose is to prove existence and regularity results for such quasi- linear elliptic systems with quadratic growth in the Du terms. The coefficients of the second order derivative terms are the same in each equation of this system. The details are worked out for a two dimensional system of the form: \(u=(u_ 1,u_ 2)\), \(Au_ 1=H_ 1(x,u,Du)+f_ 1\) in \({\mathcal O}\), \(Au_ 2=H_ 2(x,u,Du)+f_ 2, u_ 1,u_ 2=0\) on \(\partial {\mathcal O}\), with quadratic growth of \(H_ i\) with respect to Du. The quadratic growth of \(H_ 1\) with respect to \(Du_ 2\) and \(H_ 2\) with respect to \(Du_ 1\) is assumed sufficiently small to obtain weak existence and regularity results. The authors show that every solution u in \(L^{\infty}\cap H^ 1_{loc}({\mathcal O},{\mathbb{R}}^ n)\) is HĂ¶lder continuous in \({\mathcal O}\). For the existence of weak solutions the \(H_ i\) functions are approximated and uniform \(L^{\infty}\), \(H^ 1\), \(C^{\alpha}\) estimates are obtained for the approximate solutions. The authors give two alternative ways of justifying the passage to the limit without using \(C^{\alpha}\)-a priori-estimates. The techniques involve a clever use of Fatou’s lemma and Gehring’s lemma. The authors also give the stochastic background and explanation of Nash equilibrium for the game.

Reviewer: S.Lenhart

##### MSC:

91A60 | Probabilistic games; gambling |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |