Shape optimization for eigenvalues of higher order elliptic operators

形状优化是几何分析中一个快速发展的领域。在形式上，形状优化问题可以表述为：给定一个可容许形状集合a和一个定域泛函F，我们寻找a中使F在a中最小的一个元素，一个经典的例子是等周不等式，它是最古老的最优形状问题之一。在这种情况下，A是所有体积固定的开集的集合，F是一个定义域的周长。如果涉及到偏微分方程的解，事情就变得更复杂了。在这种情况下，泛函F依赖于定义域D和给定偏微分方程在D上的解u，我们感兴趣的是理解这类问题。我们被引导到以下问题：a)是否存在最优域？b)最优域是否正则？c)如果是，能否制定最优性的必要条件？最优域是否唯一？对于二阶椭圆型偏微分方程的问题，有几种已知的策略可以帮助回答上述问题。这些策略主要基于最大原则、放大技术或对称化论证（如Alt和Caffarelli）。不幸的是，这些策略最多只能在二阶偏微分方程出现时才有效。然而，我们对涉及高阶椭圆型偏微分方程的问题感兴趣。因此，前面提到的策略不适用。在这个项目中，我们集中研究了两个特殊的域泛函，其中涉及到四阶偏微分方程的解。即，固定板的主频率和屈曲载荷。这个项目的核心是推进这两个问题的最新进展。由于前面提到的策略不适用，挑战是开发新的方法来分析我们的最优形状问题。对于屈曲载荷，我们最近在回答问题a)和c)方面取得了重大进展。我们将尝试扩展我们的结果，并将其转移到基频问题。目前，对于屈曲载荷的最小域或夹紧板的主频率，还没有已知的规律性结果。受二阶椭圆问题的Alt-Caffarelli方法的启发，本课题的目的之一是寻找一种利用关联解的正则性来获得最优域的正则性的方法。

Shape optimization is a quickly growing field within geometric analysis. Formally, shape optimization problems are formulated as follows: given a set A of admissible shapes and a domain functional F, we look for an element of A which minimizes F in A. A classical example is the isoperimetric inequality, which is one of the oldest optimal shape problems. In this case, A is the set of all open sets with fixed volume and F is the perimeter of a domain. Things become more complex, if solutions of partial differential equations are involved. In this case, the functional F depends on a domain D and on the solution u of a given partial differential equation on D. Our interest is to understand problems of this type. We are led to the following question: a) Does there exist an optimal domain?b) Is the optimal domain regular?c) If it is, can we formulate necessary conditons of optimality? Is the optimal domain unique?For problems concerning elliptic partial differential equations of second order, there are several strategies known which help answering the above questions. These strategies are mainly based on the maximum principle, blow-up techniques or symmetrizations arguments (see e.g. Alt and Caffarelli). Unfortunately, these strategies only work if at most second order partial differential equations occur. However, we are interested in problems in which elliptic partial differential equations of higher order are involved. Hence, the previously mentioned strategies are not applicable. In this project, we concentrate on two special domain functionals in which solutions of fourth order partial differential equations are involved. Namely, the principal frequency and the buckling load of a clamped plate. Core of this project is to advance the recent progress on these two problems. Since the previously mentioned strategies are not applicable, the challange is to develop new methods to analyze our optimal shape problems. For the buckling load, we recently gain a significant progress on answering the questions a) and c). We will try to expand our results und transfer it to the fundamental frequency problem. Currently, for a minimizing domain for the buckling load or the principal frequency of a clamped plate, there are no regularity results known. Inspired by the Alt-Caffarelli approach for second order elliptic problems, one aim of this project is to find a method to get regularity of the optimal domain using the regularity of the associated solution.