Optimal shape of a domain which minimizes the first buckling eigenvalue

• Optimale Gestalt eines Gebiets, welches den ersten gebeulten Eigenwert minimiert

Knappmann, Kathrin; Wagner, Alfred (Thesis advisor)

Aachen (2014)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2014

Abstract

The present thesis is concerned with the question which domain minimizes the buckling load of a clamped plate among all domains of given measure. We consider a large ball and restrict ourselves to look for a domain minimizing the buckling load among all open subsets of this ball, which fulfil the mass constraint. The corresponding variational problem is a minimizing problem in a suitable function space. The mass constraint is formulated as a side condition. Instead of treating this problem, we follow an idea of H. W. Alt, L. A. Caffarelli and A. Friedman and introduce a penalization term. In this way, we disregard the mass constraint as a side condition and obtain a penalized variational problem without any constraints. Consequently, we may allow non-volume preserving perturbations. Applying direct methods, we prove the existence of a solution for the penalized problem. These solutions solve the buckled plate equation in their supports’ interior. Using a technique of C. B. Morrey, we show that the first order derivatives of each minimizer are Hölder continuous. Subsequently, we establish a bound on the Laplacian of every minimizer and combine ideas of J. Frehse, L. A. Caffarelli and A. Friedman to extend this bound on each second order derivative of a minimizer. In this way, we obtain the Lipschitz continuity of the first order derivatives of each minimizer. Moreover, we find that the bound on the second order derivatives of a minimizer is uniform close to the free boundary. Consequently, we show that the penalized problem and the original problem can be treated as equivalent, provided the penalization parameter is chosen smaller than a critical value. Hence, we obtain a solution for the original problem. For each solution, the interior of the solution’s support generates an optimal domain for minimizing the buckling eigenvalue among all subsets of the embracing ball, which fulfil the volume condition. The solutions themselves are first buckling eigenfunctions. Assuming that a doubling property is satisfied, we show that the solutions do not degenerate along the free boundary. This nondegeneracy property implies a lower bound on the density of the free boundary. As a consequence of this density bound, we obtain that an optimal domain’s boundary is a nullset with respect to the n-dimensional Lebesgue measure and prove qualitative properties of an optimal shape.