Shape sensitivity analysis of the eigenvalues of polyharmonic operators and elliptic systems
Davide Buoso · Padua@research (University of Padova) · 2015
In this thesis, we study the dependence of the eigenvalues of elliptic partial dierential operators upon domain perturbations in the N-dimensional space. Namely, we prove analyticity results for the eigenvalues of polyharmonic operators and elliptic systems of second order partial differential equations, and we apply them to certain shape optimization problems. On the other hand, we also prove spectral stability estimates for general elliptic systems of partial differential equations of higher order. In order to prove analyticity, we use a general technique developed by Lamberti and Lanza de Cristoforis, and we obtain Hadamard-type formulas which are used to provide a characterization of critical domains under volume constraint. As for stability estimates of the eigenvalues, we prove indeed Lipschitz continuity results with respect to the atlas distance, the Hausdor distance and the Lebesgue measure. We adapt the arguments used by Burenkov and Lamberti for elliptic operators to the case of general elliptic systems of partial differential equations. The thesis is organized as follows. Chapter 1 is dedicated to some preliminaries. In Chapter 2 we consider the biharmonic operator under different boundary conditions, namely Dirichlet, Neumann, intermediate and Steklov. For all these cases we show analytic dependence of the eigenvalues upon the domain and compute Hadamard-type formulas, which will be used to provide a characterization of critical domains for the elementary symmetric functions of the eigenvalues under volume constraint. Then we prove that balls are critical domains for such functions of the eigenvalues of all these problems under volume constraint. Regarding the Steklov problem, we also prove that the ball is a maximizer of the fundamental tone among all bounded open sets of given measure. In Chapter 3 we consider the Dirichlet eigenvalue problem for general polyharmonic operators. As in Chapter 2, we prove analyticity of the elementary symmetric functions of the eigenvalues providing Hadamard-type formulas, and we give a characterization of critical domains under volume constraint. Then we show that for all the polyharmonic operators the ball is a critical domain. Chapter 4 is devoted to the stability estimates of the eigenvalues of elliptic systems of partial differential equations with Dirichlet and Neumann boundary conditions. Adapting the arguments used by Burenkov and Lamberti for elliptic operators, we can prove estimates via the atlas distance, the lower Hausdor-Pompeiu deviation and the Lebesgue measure. In Chapter 5 we prove analyticity, Hadamard-type formulas and criticality conditions for second order elliptic systems under Dirichlet and Neumann boundary conditions. We also show that, if the system is rotation invariant, then balls are critical domains under volume constraint. Finally, in Chapter 6 we consider the Reissner-Mindlin problem for the vibration of a clamped plate. We first prove estimates similar to those of Chapter 4, which are independent of the thickness of the plate. Then we prove analyticity and Hadamard-type formulas for the elementary symmetric functions of the eigenvalues, which are used to provide a characterization of criticality. Then, after proving that the Reissner-Mindlin system is rotation invariant, we show that balls are critical domains under volume constraint.