论文标题
半线性椭圆形偏微分方程的Dirichlet类型能量的形状优化
Shape optimization of a Dirichlet type energy for semilinear elliptic partial differential equations
论文作者
论文摘要
在体积约束下，将所谓的“ Dirichlet Energy”最小化是形状优化的标准问题，现在已被充分理解。本文致力于该问题的原型非线性版本，其中旨在最大程度地减少涉及域中针对该域的半线性椭圆PDE解决方案在体积约束下涉及到该域的能量。与此问题的标准版本的主要区别之一是基于以下事实：最小化的标准并不是能量的最低限度，因此无法使用分析此问题的大多数常用工具。通过使用此问题的放松版本，我们首先证明了有关问题参数的几个假设的最佳形状存在。然后，我们分析球的稳定性，预计将是解决形状优化问题的良好候选者，当相关PDE的系数径向对称时。
Minimizing the so-called "Dirichlet energy" with respect to the domain under a volume constraint is a standard problem in shape optimization which is now well understood. This article is devoted to a prototypal non-linear version of the problem, where one aims at minimizing a Dirichlet-type energy involving the solution to a semilinear elliptic PDE with respect to the domain, under a volume constraint. One of the main differences with the standard version of this problem rests upon the fact that the criterion to minimize does not write as the minimum of an energy, and thus most of the usual tools to analyze this problem cannot be used. By using a relaxed version of this problem, we first prove the existence of optimal shapes under several assumptions on the problem parameters. We then analyze the stability of the ball, expected to be a good candidate for solving the shape optimization problem, when the coefficients of the involved PDE are radially symmetric.
