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On Finite Element Methods for 2nd order (semi–) periodic Eigenvalue Problems

De Schepper, H. (2000)

Serdica Mathematical Journal

We deal with a class of elliptic eigenvalue problems (EVPs) on a rectangle Ω ⊂ R^2 , with periodic or semi–periodic boundary conditions (BCs) on ∂Ω. First, for both types of EVPs, we pass to a proper variational formulation which is shown to fit into the general framework of abstract EVPs for symmetric, bounded, strongly coercive bilinear forms in Hilbert spaces, see, e.g., [13, §6.2]. Next, we consider finite element methods (FEMs) without and with numerical quadrature. The aim of the paper is...

On Fredholm alternative for certain quasilinear boundary value problems

Pavel Drábek (2002)

Mathematica Bohemica

We study the Dirichlet boundary value problem for the p -Laplacian of the form - Δ p u - λ 1 | u | p - 2 u = f in Ω , u = 0 on Ω , where Ω N is a bounded domain with smooth boundary Ω , N 1 , p > 1 , f C ( Ω ¯ ) and λ 1 > 0 is the first eigenvalue of Δ p . We study the geometry of the energy functional E p ( u ) = 1 p Ω | u | p - λ 1 p Ω | u | p - Ω f u and show the difference between the case 1 < p < 2 and the case p > 2 . We also give the characterization of the right hand sides f for which the above Dirichlet problem is solvable and has multiple solutions.

On homogeneizatìon problems for the Laplace operator in partially perforated domains with Neumann's condition on the boundary of cavities.

Olga A. Oleinik, Tatiana Shaposhnikova (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper the problem of homogeneization for the Laplace operator in partially perforated domains with small cavities and the Neumann boundary conditions on the boundary of cavities is studied. The corresponding spectral problem is also considered.

On m -sectorial Schrödinger-type operators with singular potentials on manifolds of bounded geometry

Ognjen Milatovic (2004)

Commentationes Mathematicae Universitatis Carolinae

We consider a Schrödinger-type differential expression H V = * + V , where is a C -bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry ( M , g ) with metric g and positive C -bounded measure d μ , and V is a locally integrable section of the bundle of endomorphisms of E . We give a sufficient condition for m -sectoriality of a realization of H V in L 2 ( E ) . In the proof we use generalized Kato’s inequality as well as a result on the positivity of u L 2 ( M ) satisfying the...

On nodal sets and nodal domains on S 2 and 2

Alexandre Eremenko, Dmitry Jakobson, Nikolai Nadirashvili (2007)

Annales de l’institut Fourier

We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S 2 . We also construct a solution of the equation Δ u = u in 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.

Currently displaying 21 – 40 of 161