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: 18/01/2010-17/05/2013Researchers:
Pedro Freitas (PR)
Jiri Lipovsky (from January 2012)
Isabel Salavessa
Petr Siegl (from January 2012)
Publications within the scope of this project:
(for other relevant publications by the researchers involved in the project, see the respective homepages)
Published
44.
F. Morgan and
I. Salavessa
The isoperimetric problem in higher codimension
Manuscripta Math.
142 (2013), 369-382.
43.
S. Haeseler, M. Keller and
R.K. Wojciechowski
Volume growth and bounds for the essential spectrum for Dirichlet forms
J. London Math. Soc.
88 (2013), 883-898.
42.
C.J.S. Alves and
P.R.S. Antunes
The method of fundamental solutions applied to some inverse eigenproblems
SIAM J. Sci. Comput. 35 (2013), A1689-A1708.
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
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