Eigenvalues of the Laplacian: analytic, geometric and computational aspects
(ptdc/mat/101007/2008)

Description:

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.

Time span: 18/01/2010-17/01/2013

Funding institution:  

Researchers:

Publications within the scope of this project:
(for other relevant publications by the researchers involved in the project, see the respective homepages)

 Preprints

    29. J. Kennedy
          Closed nodal surfaces for simply connected domains in higher dimensions
          Indiana Univ. Math. J..

    28. D. Borisov
          On a PT -symmetric waveguide with  a pair of small holes
          Proc. Steklov Inst. Math. 18 (2012).

    27. D. Borisov and K. Pankrashkin
          Gaps opening and splitting of the zone edges for waveguides coupled by a periodic system of small windows
          Math. Notes.

    26. D. Borisov and K. Pankrashkin
          On extrema of band functions in periodic waveguides
          Funct. Anal. Appl..

    25. P.R.S. Antunes and F. Gazzola
          Convex shape optimization for the least biharmonic Steklov eigenvalue
          ESAIM: Control, Optimisation and Calculus of Variations.

    24. I. Salavessa
         Stable 3-spheres in C3
          J. Math. Research.

    23. P.R.S. Antunes and P. Freitas
          Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians
          J. Opt. Theory Appl. .

    22. D. Borisov and P. Freitas
          Asymptotics for the expected lifetime of Brownian motion on thin domains in Rn
          J. Theoret. Probab..

 Published

    21. D. Borisov and G. Cardone
          Planar waveguide with "twisted" boundary conditions: small width
          J. Math. Phys. 53 (2012), 023503.

    20. D. Borisov and P. Freitas
          Eigenvalue asymptotics for almost flat compact hypersurfaces
          Dokl. Akad. Nauk. 442 (2012), 151-155; translation in Dokl. Math. 85 (2012), 18-22.

    19. D. Borisov and G. Cardone
          Planar waveguide with "twisted" boundary conditions: discrete spectrum
          J. Math. Phys. 52 (2011), 123513.

    18. D. Borisov
          On spectrum of two-dimensional periodic operator with small localized perturbation
         Izvestia Math. 75 (2011), 471-505.

    17. J. Kennedy
          The nodal line of the second eigenfunction of the Robin Laplacian in R2 can be closed
          J. Differential Equations 251 (2011), 3606-3624.

    16. P.R.S. Antunes
         Numerical calculation of eigensolutions of 3D shapes using the Method of Fundamental Solutions
         Numer. Methods Partial Differential Equations 27 (2011), 1525-2550.

    15. B. Brandolini, P. Freitas, C. Nitsch and C. Trombetti
          Sharp estimates and saturation phenomena for a nonlocal eigenvalue problem
          Adv. Math. 228 (2011), 2352-2365.

    14. D. Borisov and G. Cardone
          Complete asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in thin three-dimensional rods
         ESAIM: Control, Optimisation and Calculus of Variations 17 (2011), 887-908.

    13. 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 64 (2011), 163-179, Birkhaeuser.

    12. P.R.S. Antunes
          On the buckling eigenvalue problem
         J. Phys. A 44 (2011), 215205.

    11. P.R.S. Antunes and A. Henrot
          On the range of the first two Dirichlet and nontrivial Neumann eigenvalues of the Laplacian
          Proc. Royal Soc. A Math. Phys. Eng. Sci. 467 (2011), 1577-1603.

    10. P.R.S. Antunes and P. Freitas
          On the inverse spectral problem for Euclidean triangles
          Proc. Royal Soc. A Math. Phys. Eng. Sci. 467 (2011), 1546-1562.

      9. D. Borisov , R. Bunoiu and G. Cardone
          On a waveguide with infinite number of small windows
          Compt. Rend. Math. 349 (2011), 53-56.

      8. D. Borisov and I. Veselic'
          Low lying spectrum of weak-disorder quantum waveguides
          J. Statistical Phys. 142 (2011), 58-77.

      7. P. Freitas and I. Salavessa
         A spectral Bernstein theorem
         Ann. Mat. Pura Appl. 190 (2011), 77-90.

      6. D. Borisov, R. Bunoiu, and G. Cardone
          On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition
          Ann. Henri Poincaré 11 (2010), 1591-1627.

      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 and 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 and 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 and P. Freitas
         Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in Rd
         J. Funct. Anal. 258 (2010), 893-912.

      1. D. Borisov and P. Freitas
         Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation
          J. Differential Equations 247 (2009), 3028-3039.


Group of Mathematical Physics, University of Lisbon
Complexo Interdisciplinar, Av. Prof. Gama Pinto 2,
P-1649-003 Lisboa, Portugal