# Valores próprios do Laplaciano: aspectos analíticos, geométricos e computacionais

PTDC/MAT/101007/2008

## Descrição

The purpose of the project is to combine analytic, geometric and computational aspects to develop the theory of eigenvalues of the Laplacian and related operators. The emphasis will be on the study of isoperimetric relations between spectral and geometric quantities and on the approximation of eigenvalues from numerical and analytic perspectives.

The host institution is the Group of Mathematical Physics of the University of Lisbon. This is a research centre in Mathematics funded by the Portuguese Science Foundation (FCT) which has always been awarded the highest possible classification in all international evaluations carried out by FCT; in the latest of these (2008) only 6 research units out of a total universe of 20 Maths Centres in the whole country received this classification.

The project includes funding for **two postdoctoral
positions** starting before the end of 2010.

Time span: 2010-01-14 / 2013-01-13

Funding institution: FCT.

## Investigadores

- Clara Aldana
- Carlos Alves
- Pedro Antunes
- Denis Borisov
- Pedro Freitas (IR)
- James Kennedy
- Isabel Salavessa
- Radoslaw Wojciechowski

## Publicações no âmbito deste projecto

(poderá ver informação sobre outras publicações dos investigadores envolvidos no projecto, consultando as respectivas páginas)

### Pré-publicações

- 11. P.R.S. Antunes e P. Freitas
- On the inverse spectral problem for Euclidean triangles,

*Proc. Royal Soc. A Math. Phys. Eng. Sci.*. - 10. D. Borisov
- On spectrum of two-dimensional periodic operator with small
localized perturbation,

*Izvestia Math.*. - 9. R. Wojciechowski
- Stochastically incomplete manifolds and graphs,

*Boundaries and Spectra of Random Walks*

(D. Lenz, F. Sobieczky and W. Woess, ed.), Proceedings, Graz — St. Kathrein 2009; Progress in Probability, Birkhaeuser. - 8. D. Borisov e G. Cardone
- Complete asymptotic expansions for the eigenvalues of the
Dirichlet Laplacian in thin three-dimensional rods,

*ESAIM: Control, Optimisation and Calculus of Variations*. - 7. P. Freitas e I. Salavessa
- A spectral Bernstein theorem,

*Ann. Mat. Pura Appl.*. - 6. P. R. S. Antunes
- Numerical calculation of eigensolutions of 3D shapes using the
Method of Fundamental Solutions,

*Numer. Methods Partial Differential Equations*.

### Artigos publicados

- 5. I. Salavessa
- Stability of submanifolds with parallel mean curvature
in calibrated manifolds,

*Bull. Brazilian Math. Soc. (NS)***41**(2010), 495-530. - 4. P. Freitas e B. Siudeja
- Bounds for the first Dirichlet eigenvalue of triangles and
quadrilaterals,

*ESAIM: Control, Optimisation and Calculus of Variations***32**(2010), 189-200. - 3. P. R. S. Antunes e S.S. Valtchev
- A meshfree numerical method for acoustic wave propagation problems
in planar domains with corners and cracks,

*J. Comp. Appl. Math.***234**(2010), 2646-2662. - 2. D. Borisov e P. Freitas
- Asymptotics of Dirichlet eigenvalues and eigenfunctions of the
Laplacian on thin domains in R
^{d},

*J. Funct. Anal.***258**(2010), 893-912. - 1. D. Borisov e P. Freitas
- Eigenvalue asymptotics, inverse problems and a trace formula for
the linear damped wave equation,

*J. Differential Equations***247**(2009), 3028-3039.