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: 18/01/2010-17/05/2013

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)

 Published

    41. D. Borisov and P. Freitas
          On the spectrum of deformations of compact double-sided flat hypersurfaces

          Anal. PDE 6 (2013), 1051-1088.

    40. M. Keller, D. Lenz and R.K. Wojciechowski
          Volume growth, spectrum and stochastic completeness of infinite graphs

          Math. Z. 274 (2013), 905-932.

    39. J. Kennedy
          Closed nodal surfaces for simply connected domains in higher dimensions
          Indiana Univ. Math. J. 62 (2013), 785-798.

    38. D. Borisov
          On a PT-symmetric waveguide with a pair of small holes
          Proc. Steklov Inst. Math. 281 (2013), S5-S21.

    37. 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 93 (2013), 665-683.

    36. D. Borisov and K. Pankrashkin
          On extrema of band functions in periodic waveguides

          Funct. Anal. Appl. 47 (2013), 283-240.

    35. I. Salavessa
         Stable 3-spheres in C3

          J. Math. Research 4 (2012), 34-44.

    34. P. Albin, C.L. Aldana and F. Rochon
          Ricci flow and the determinant of the Laplacian on non-compact surfaces.

          Commun. Partial Diff. Eq. 38 (2013), 711-749

    33. D. Bucur and P. Freitas
          Asymptotic behaviour of optimal spectral planar domains with fixed perimeter

          J. Math. Phys. 54 (2013), 053504

    32. P.R.S. Antunes, P. Freitas and J. Kennedy
          Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian
          ESAIM: Control, Optimisation and Calculus of Variations 19 (2013), 438-459.

    31. P.R.S. Antunes and F. Gazzola
          Convex shape optimization for the least biharmonic Steklov eigenvalue

          ESAIM: Control, Optimisation and Calculus of Variations19 (2013), 385-403.

    30. D. Borisov and P. Freitas
          Asymptotics for the expected lifetime of Brownian motion on thin domains in Rn

          J. Theoret. Probab. 26 (2013), 284-309.

    29. P.R.S. Antunes
          Optimization of sums and quotients of Dirichlet–Laplacian eigenvalues

          Appl. Math. Comput. 219 (2013), 4239-4254.

    28. P.R.S. Antunes and P. Freitas
          Optimal spectral rectangles and lattice ellipses

          Proc. Royal Soc. A Math. Phys. Eng. Sci. 469 (2013), 20120492.

    27. D. Krejcirik and P. Siegl
          On the metric operator for the imaginary cubic oscillator

          Phys. Rev. D 86 (2012), 121702(R)

    26. C.L. Aldana
          Determinants of Laplacians on non-compact surfaces

          Contemp. Math. 584 (2012), 223-236

    25. D. Kochan, D. Krejcirik, R. Novak, and P. Siegl
          The Pauli equation with complex boundary conditions

         J. Phys. A 45 (2012), 444019.

    24. S. Haeseler, M. Keller, D. Lenz and R.K. Wojciechowski
          Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

          J. Spectr. Theory 2 (2012), 397-432.

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

          J. Opt. Theory Appl. 154 (2012), 235-257.

    22. D. Borisov and G. Cardone
          Planar waveguide with "twisted" boundary conditions: small width

          J. Math. Phys. 53 (2012), 023503.

    21. 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.

    20. D. Borisov and G. Cardone
          Planar waveguide with "twisted" boundary conditions: discrete spectrum

          J. Math. Phys. 52 (2011), 123513.

    19. D. Borisov
          On spectrum of two-dimensional periodic operator with small localized perturbation

          Izvestia Math. 75 (2011), 471-505.

    18. 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.

    17. 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.

    16. 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.

    15. 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.

    14. R.K. 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.

    13. P.R.S. Antunes
          On the buckling eigenvalue problem

         J. Phys. A 44 (2011), 215205.

    12. C.J.S. Alves
          Coupling MFS with BEM approximation

         Conf. Boundary Integral Methods D. Lesnic, Ed. (2011).

    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.


e-mail:psfreitas ( a@t )fc.ul.pt

Group of Mathematical Physics - University of Lisbon
Department of Mathematics
Faculty of Sciences
Campo Grande, Edifício C6
P-1749-016 Lisboa, Portugal