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Shape sensitivity analysis of the eigenvalues of polyharmonic operators and elliptic systems
Full text linkIn 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
Sharp inequalities and asymptotics for polyharmonic eigenvalues
No full textWe study eigenvalues of general scalar Dirichlet polyharmonic problems in domains in. We first prove a number of inequalities satisfied by the eigenvalues on general domains, depending on the relations between the orders of the operators involved. We then obtain several estimates for these eigenvalues, yielding their growth as a function of these orders. For the problem in the ball we derive the general form of eigenfunctions together with the equations satisfied by the corresponding eigenvalues, and obtain several bounds for the first eigenvalue. In the case of the polyharmonic operator of order 2m we derive precise bounds yielding the first two terms in the asymptotic expansion for the first normalised eigenvalue as m grows to infinity. These results allow us to obtain the order of growth for the polyharmonic eigenvalue on general domains
Analyticity and criticality results for the eigenvalues of the biharmonic operator
No full textWe consider the eigenvalues of the biharmonic operator subject to several homogeneous boundary conditions (Dirichlet, Neumann, Navier, Steklov). We show that simple eigenvalues and elementary symmetric functions of multiple eigenvalues are real analytic, and provide Hadamard-type formulas for the corresponding shape derivatives. After recalling the known results in shape optimization, we prove that balls are always critical domains under volume constraint
A few shape optimization results for a biharmonic Steklov problem
Full text linkWe derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue
The Bilaplacian with Robin Boundary Conditions
Full text linkWe 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
On a classical spectral optimization problem in linear elasticity
No full textWe consider a classical shape optimization problem for the eigenvalues of elliptic operators with homogeneous boundary conditions on domains in the N-dimensional Euclidean space. We survey recent results concerning the analytic dependence of the elementary symmetric functions of the eigenvalues upon domain perturbation and the role of balls as critical points of such functions subject to volume constraint. Our discussion concerns Dirichlet and buckling-type problems for polyharmonic operators, the Neumann and the intermediate problems for the biharmonic operator, the Lame' and the Reissner-Mindlin systems
Shape Sensitivity Analysis of the Eigenvalues of the Reissner--Mindlin System
Full text linkWe 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
Shape deformation for vibrating hinged plates
Full text linkWe consider the biharmonic operator subject to homogeneous intermediate boundary conditions of Steklov-type. We prove an analyticity result for the dependence of the eigenvalues upon domain perturbation and compute the appropriate Hadamard-type formulas for the shape derivatives. Finally, we prove that balls are critical domains for the symmetric functions of multiple eigenvalues subject to volume constraint
Eigenvalues of polyharmonic operators on variable domains
Full text linkWe consider a class of eigenvalue problems for polyharmonic operators, including Dirichlet and buckling-type eigenvalue problems. We prove an analyticity result for the dependence of the symmetric functions of the eigenvalues upon domain perturbations and compute Hadamard-type formulas for the Frechét differentials. We also consider isovolumetric domain perturbations and characterize the corresponding critical domains for the symmetric functions of the eigenvalues. Finally, we prove that balls are critical domains
On the variation of longitudinal and torsional frequencies in a partially hinged rectangular plate
Full text linkWe consider a partially hinged rectangular plate and its normal modes. There are two families of modes, longitudinal and torsional. We study the variation of the corresponding eigenvalues under domain deformations. We investigate the possibility of finding a shape functional able to quantify the torsional instability of the plate, namely how prone is the plate to transform longitudinal oscillations into torsional ones. This functional should obey several rules coming from both theoretical and practical evidences. We show that a simple functional obeying all the required rules does not exist and that the functionals available in literature are not reliable
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