31 research outputs found
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 MacIntyreâs 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 of
the eigenvalues of various biharmonic operators, with a second term in the
expected power of . 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 . We also prove a
Berezin-Li-Yau inequality for domains contained in the hemisphere . 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 analiticitĂ 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 stabilitĂ spettrale per sistemi ellittici generali
di equazioni alle derivate parziali di ordine superiore. Per dimostrare
l'analiticitĂ 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 stabilitĂ degli autovalori, dimostriamo
risultati di lipschitzianitĂ 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'analiticitĂ 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 stabilitĂ 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 analiticitĂ , formule alla
Hadamard e condizioni di criticitĂ 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'analiticitĂ e formule alla Hadamard
per le funzioni elementari simmetriche degli autovalori, che vengono usate
per fornire una caratterizzazione di criticitĂ ; 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