Shape optimization for eigenvalues of higher order elliptic operators

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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.