随机摄动椭圆问题的最优形状
Optimal Shape for Elliptic Problems with Random Perturbations
摘要：
本文分析了状态方程为$\{{array}{ll} -div \big(a(x)Du\big)=f\qquad\hbox{in}D \hbox{boundary conditions on}\partial D. {array}.$的形状最优化问题的松弛形式。新的事实是，$f$项只在随机扰动$ξ(x,ω)$下才是已知的。我们的目标是找到一个最佳的系数$a(x)$，满足通常的约束$α\le a\leβ$和$\displaystyle\int_D a(x) dx\le m$，最小化的费用函数的形式$$\int_Ω\int_Dj\big(x,ω,u_a(x,ω)\big) dx dP(ω).$$的一些数值例子中示出的最后一节，强调与无扰动的情况下的差异。
英文摘要：
In this paper we analyze the relaxed form of a shape optimization problem with state equation $\{{array}{ll} -div \big(a(x)Du\big)=f\qquad\hbox{in}D \hbox{boundary conditions on}\partial D. {array}.$ The new fact is that the term $f$ is only known up to a random perturbation $ξ(x,ω)$. The goal is to find an optimal coefficient $a(x)$, fulfilling the usual constraints $α\le a\leβ$ and $\displaystyle\int_D a(x) dx\le m$, which minimizes a cost function of the form $$\int_Ω\int_Dj\big(x,ω,u_a(x,ω)\big) dx dP(ω).$$ Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.
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