一类椭圆型随机偏微分方程弱解的存在性

一类椭圆型随机偏微分方程弱解的存在性

Existence of Weak Solutions to a Class of Elliptic Stochastic

Partial Differential Equations

摘要/Abstract

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数学物理学报

冉启康

海财经大学应用数学系上海 200433

收稿日期:2006-06-18 修回日期:2007-07-17 出版日期:2008-04-25 发布日期:2008-04-25
通讯作者: 冉启康
基金资助:
国家自然科学基金(10371021)资助

Ran Qikang

Department of Applied Mathematics, Shanghai University of Finance and Economics,Shanghai 200433

Received:2006-06-18 Revised:2007-07-17 Online:2008-04-25 Published:2008-04-25
Contact: Ran Qikang

摘要： 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

{\begin{cases} - d i v A (x, ω, u, \nabla u) = f (x, ω, u), & (x, ω) \in D \times Ω, \\ u = 0, & (x, ω) \in \partial D \times Ω, \end{cases}

$\left\{\begin{array}{ll} -{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &(x,\omega)\in D\times \Omega,\\ u=0, &(x,\omega)\in \partial D\times \Omega, \end{array}\right.$ 其中, div与

\nabla

$\nabla$ 表示仅对

x

$x$ 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.

关键词: 非线性椭圆随机偏微分方程, 弱解, Leray-Schauder连续方法

Abstract: 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:

{\begin{cases} - d i v A (x, ω, u, \nabla u) = f (x, ω, u), & (x, ω) \in D \times Ω, \\ u = 0, & (x, ω) \in \partial D \times Ω, \end{cases}

$\left\{ \begin{array}{ll} -{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &(x,\omega)\in D\times \Omega,\\ u=0, &(x,\omega)\in \partial D\times \Omega, \end{array}\right.$
where by div and

\nabla

$\nabla$ the authors denote differentiation with respect to

x

$x$ only. First, the authors
introduce the concept of the weak solution, then the authors transform the stochastic problem into a deterministic
one in high-dimensions. Finally, the authors prove the existence of weak solutions.

Key words: Nonlinear elliptic stochastic partial differential equations, Weak solutions,
Leray-Schauder continuation method

中图分类号:

35J65

冉启康. 一类椭圆型随机偏微分方程弱解的存在性[J]. 数学物理学报, 2008, 28(2): 320-328.

Ran Qikang.

Existence of Weak Solutions to a Class of Elliptic Stochastic

Partial Differential Equations

[J]. Acta mathematica scientia,Series A, 2008, 28(2): 320-328.

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