一类椭圆型随机偏微分方程弱解的存在性

设$D$是$R^N$ ($N>1$)中有界开集,$(\Omega, {\cal F}, P)$是一个完备的概率空间.该文研究了下列随机边值问题弱解的存在性问题\\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.\]其中, div与 $\nabla $ 表示仅对 $x$求微分. 首先,作者引入了弱解的概念; 然后,作者转化随机问题为高维确定性问题;最后,作者证明了该问题弱解的存在性.

Existence of Weak Solutions to a Class of Elliptic Stochastic Partial Differential Equations

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.