Shape differentiability of the eigenvalues of elliptic systems

We consider second order elliptic systems of partial differential equations subject to Dirichlet and Neumann boundary conditions. We prove analyticity of the elementary symmetric functions of the eigenvalues, and compute Hadamard-type formulas for such functions. Then we provide a characterization of criticality of the domain under volume constraint, and prove that if the system is rotation invariant, then balls are critical domains for all those functions.Comment: Submitted for the IMSE 2014 Conference Proceeding

On the eigenvalues of a biharmonic Steklov problem

We consider an eigenvalue problem for the biharmonic operator with Steklov-type boundary conditions. We obtain it as a limiting Neumann problem for the biharmonic operator in a process of mass concentration at the boundary. We study the dependence of the spectrum upon the domain. We show analyticity of the symmetric functions of the eigenvalues under isovolumetric perturbations and prove that balls are critical points for such functions under measure constraint. Moreover, we show that the ball is a maximizer for the first positive eigenvalue among those domains with a prescribed fixed measure.Comment: This paper will appear in the proceedings of the IMSE 2014 Conferenc

Alasdair MacIntyre’s Contribution to Marxism: A Road not Taken

This essay questions, through a critique of his reading of classical Marxism, the path taken by Alasdair MacIntyre since his break with the Marxist Left in the 1960s. It argues that MacIntyre was uncharitable in his criticisms of Marxism, or at least in his conflation of the most powerful aspects of the classical Marxist tradition with the crudities of Kautskyian and Stalinist materialism. Contra MacIntyre, this essay locates in the writings of the revolutionary Left which briefly flourished up to and just after the Russian Revolution a rich source of dialectical thinking on the relationship between structure and agency that escapes the twin errors of crude materialism or political voluntarism. Moreover, it suggests that by reaching back to themes reminiscent of the young Marx this tradition laid the basis for a renewed ethical Marxism, and that in his youth MacIntyre pointed to the realisation of this project

A Minimaxmax Problem for Improving the Torsional Stability of Rectangular Plates

We use a gap function in order to compare the torsional performances of different reinforced plates under the action of external forces. Then, we address a shape optimization problem, whose target is to minimize the torsional displacements of the plate: this leads us to set up a minimaxmax problem, which includes a new kind of worst-case optimization. Two kinds of reinforcements are considered: one aims at strengthening the plate, the other aims at weakening the action of the external forces. For both of them, we study the existence of optima within suitable classes of external forces and reinforcements. Our results are complemented with numerical experiments and with a number of open problems and conjectures

Semiclassical bounds for spectra of biharmonic operators

We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ of the eigenvalues of various biharmonic operators, with a second term in the expected power of $z$. The method we discuss makes use of the averaged variational principle (AVP), and yields two-sided bounds for individual eigenvalues, which are semiclassically sharp. The AVP also yields comparisons with Riesz means of different operators, in particular Laplacians

The Bilaplacian with Robin Boundary Conditions

We introduce Robin boundary conditions for biharmonic operators, which are a model for elastically supported plates and are closely related to the study of spaces of traces of Sobolev functions. We study the dependence of the operator, its eigenvalues, and eigenfunctions on the Robin parameters. We show in particular that when the parameters go to plus infinity the Robin problem converges to other biharmonic problems, and we obtain estimates on the rate of divergence when the parameters go to minus infinity. We also analyze the dependence of the operator on smooth perturbations of the domain, computing the shape derivatives of the eigenvalues and giving a characterization for critical domains under volume and perimeter constraints. We include a number of open problems arising in the context of our results

Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system

We consider the eigenvalue problem for the Reissner-Mindlin system arising in the study of the free vibration modes of an elastic clamped plate. We provide quantitative estimates for the variation of the eigenvalues upon variation of the shape of the plate. We also prove analyticity results and establish Hadamard-type formulas. Finally, we address the problem of minimization of the eigenvalues in the case of isovolumetric domain perturbations. In the spirit of the Rayleigh conjecture for the biharmonic operator, we prove that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigenvalues.Comment: Preprint version of a paper accepted for publication in SIAM Journal on Mathematical Analysi

Semiclassical estimates for eigenvalue means of Laplacians on spheres

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for P\'olya-type inequalities. Finally, we provide sum rules for Laplacian eigenvalues on spheres and compact two-point homogeneous spaces

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

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.In questa tesi, studiamo la dipendenza degli autovalori di operatori differenziali ellittici da perturbazioni del dominio nello spazio N-dimensionale. In particolare, proviamo risultati di analiticità degli autovalori di operatori poliarmonici e sistemi ellittici di equazioni alle derivate parziali del secondo ordine, e li applichiamo a problemi di ottimizzazione di forma; d'altro canto, otteniamo anche stime di stabilità spettrale per sistemi ellittici generali di equazioni alle derivate parziali di ordine superiore. Per dimostrare l'analiticità usiamo una tecnica generale sviluppata da Lamberti e Lanza de Cristoforis, e otteniamo delle formule alla Hadamard che ci permettono di fornire una caratterizzazione dei domini critici sotto il vincolo di volume. Per quanto riguarda le stime di stabilità degli autovalori, dimostriamo risultati di lipschitzianità rispetto alla distanza d'atlante, alla distanza di Hausdorff e alla misura di Lebesgue, adattando gli argomenti utilizzati da Burenkov e Lamberti per operatori ellittici al caso di sistemi ellittici generali di equazioni alle derivate parziali. La tesi e organizzata come segue. Il Capitolo 1 e dedicato ad alcuni preliminari. Nel Capitolo 2 consideriamo l'operatore biarmonico con diverse condizioni al contorno, ovvero di Dirichlet, di Neumann, intermedie e di Steklov. Per tutti questi casi mostriamo la dipendenza analitica degli autovalori dal dominio e calcoliamo formule alla Hadamard, che vengono usate per formire una caratterizzazione dei domini critici per le funzioni elementari simmetriche degli autovalori sotto il vincolo di volume; a seguire proviamo che le palle sono domini critici per tali funzioni degli autovalori di tutti questi problemi sotto il vincolo di volume. Riguardo al problema di Steklov, mostriamo anche che la palla e un massimizzatore del tono fondamentale tra tutti gli aperti limitati di misura fissata. Nel Capitolo 3 consideriamo il problema agli autovalori con condizioni di Dirichlet per gli operatori poliarmonici. Come nel Capitolo 2, dimostriamo l'analiticità delle funzioni elementari simmetriche degli autovalori fornendo formule alla Hadamard, e diamo una caratterizzazione dei domini critici sotto il vincolo di volume; a seguire mostriamo che per tutti gli operatori poliarmonici la palla e un dominio critico. Il Capitolo 4 e dedicato alle stime di stabilità degli autovalori dei sistemi ellittici di equazioni alle derivate parziali con condizioni al bordo di Dirichlet e di Neumann. Adattando gli argomenti usati da Burenkov e Lamberti per operatori ellittici siamo in grado di provare stime con la distanza d'atlante, con la deviazione inferiore di Hausdorff-Pompeiu e con la misura di Lebesgue. Nel Capitolo 5 dimostriamo analiticità, formule alla Hadamard e condizioni di criticità per sistemi ellittici del secondo ordine con condizioni al bordo di Dirichlet e di Neumann. Mostriamo anche che, se il sistema e invariante per rotazioni, allora le palle sono domini critici sotto il vincolo di volume. Infine, nel Capitolo 6 consideriamo il problema di Reissner-Mindlin per la vibrazione di una piastra incastrata. Prima dimostriamo stime simili a quelle del Capitolo 4, che non dipendono dallo spessore della piastra; poi dimostriamo l'analiticità e formule alla Hadamard per le funzioni elementari simmetriche degli autovalori, che vengono usate per fornire una caratterizzazione di criticità; a seguire, dopo aver provato che il sistema di Reissner-Mindlin e invariante per rotazioni, mostriamo che le palle sono domini critici sotto il vincolo di volume