Shape sensitivity analysis of the eigenvalues of polyharmonic operators and elliptic systems

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