Uniqueness of viscosity solutions for monotone systems of fully nonlinear PDES under Dirichlet condition

Abstract

is a given function, where L2 is a bounded open set of R” and S” is the set of symmetric matrices of order n. Several authors have studied some systems of PDEs via viscosity solution methods; e.g. [l-5]. Recently, Ishii and Koike [6] have pointed out that these systems have a monotone structure and have obtained uniqueness results under the assumption of monotonicity on F. Moreover, Ishii [7] has shown that Perron’s method gives the existence result under this monotonicity hypothesis. On the other hand, it is known that, in general, boundary conditions should be considered in the viscosity sense especially in the case when F is degenerate elliptic. In the case of single PDEs Ishii [8] first pointed out this fact and studied Dirichlet boundary value problems in the viscosity sense. We also refer to [9-131. The aim of this paper is to obtain some uniqueness results for viscosity solutions of (1.1) under Dirichlet condition in the viscosity sense. By using a perturbation technique in [3] we will also weaken a uniform continuity assumption on F. This paper is organized as follows. In Section 2 we recall the definition of viscosity solutions and give an equivalent definition. In Section 3 we state our main results and give some remarks. In Section 4 we prove the main results. In the final section, as a modification of main results, we will treat a so-called switching game with Dirichlet data in the viscosity sense.