Existence of Weak Solutions to a Class of Elliptic Stochastic Partial Differential Equations
一类椭圆型随机偏微分方程弱解的存在性

In this paper the authors study of following problem: Let $D$ be a bounded open set of $R^N(N>1)$ and $(\Omega,F,P)$ is a probability space. The authors study the existence of weak solutions of the following stochastic boundary value problem:$$\left\{\begin{array}{ll}-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &(x,\omega)\in D\times \Omega,\\u=0, &(x,\omega)\in \partial D\times \Omega,\end{array}\right.$$where by div and $\nabla$ the authors denote differentiation with respect to $x$ only. First, the authorsintroduce the concept of the weak solution, then the authors transform the stochastic problem into a deterministicone in high-dimensions. Finally, the authors prove the existence of weak solutions.