Shape optimization for eigenvalues of higher order elliptic operators
Final Report Abstract
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 conditions 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. 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 concentrated on two special domain functionals in which solutions of fourth order partial differential equations are involved. Namely, the fundamental tone and the buckling load of a clamped plate. Core of this project was to advance the progress on these two problems. Since the previously mentioned strategies are not applicable, the challenge was to develop new methods to analyze our optimal shape problems. We proved the existence of an optimal domain for minimizing the buckling load among all open and bounded subset of Rn , n ≥ 2, with given measure. Thereby we followed an idea of Alt and Caffarelli and formulated the minimizing of the buckling load as a free boundary value problem with a penalization term for the measure constraint. Two different choices of penalization term were discussed. Considering the fundamental tone of a clamped plate we achieved the analog results applying exactly the same techniques as in the case of the buckling load. Moreover, we extended a two-dimensional existence result of Ashbaugh and Bucur to arbitrary dimension. Ashbaugh and Bucur proved the existence of an optimal domain for minimizing the buckling load among all plane, possibly unbounded, open subsets of R2 with given measure by applying an concentration-compactness principle. We combined their idea with the Alt-Caffarelli ansatz to focus on the eigenfunction and, thus, proved the existence of an optimal domain among all open and possibly unbounded subset of Rn with given measure. Another part of this project was a collaboration with Alexandra Gilsbach from Tokyo Institute of Technology. We examined an overdetermined Serrin-type boundary value problem for the biharmonic operator. In this problem, a solution exists for a constant overdetermining condition if the underlying set is the unit ball. We proved the existence of an open and bounded domain admitting a solution to the boundary value problem for every small perturbation of the overdetermining condition. In addition, we deduced stability estimates for the deviation of this domain from the unit ball in terms of the perturbation. Our approach is motivated by a recent result of Gilsbach and Onodera and applies a result of Ferrero, Gazzola and Weth for a fourth order Steklov problem.
Publications
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Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension
K. Stollenwerk
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“Existence and stability of solutions for a fourth order overdetermined problem”. J. Math. Anal. Appl., 505(2):Paper No. 125531, 18, 2022
Gilsbach, Alexandra & Stollenwerk, Kathrin
