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On a method of two-sided eigenvalue estimates for elliptic equations of the form A u - λ B u = 0

Karel Rektorys, Zdeněk Vospěl (1981)

Aplikace matematiky

The Collatz method of twosided eigenvalue estimates was extended by K. Rektorys in his monography Variational Methods to the case of differential equations of the form A u - λ B u = 0 with elliptic operators. This method requires to solve, successively, certain boundary value problems. In the case of partial differential equations, these problems are to be solved approximately, as a rule, and this is the source of further errors. In the work, it is shown how to estimate these additional errors, or how to avoid...

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 some new spectral estimates for Schrödinger-like operators

Daniel Levin (2006)

Open Mathematics

We prove the analog of the Cwikel-Lieb-Rozenblum estimate for a wide class of second-order elliptic operators by two different tools: Lieb-Thirring inequalities for Schrödinger operators with matrix-valued potentials and Sobolev inequalities for warped product spaces.

On the Bethe-Sommerfeld conjecture

Leonid Parnovski, Alexander V. Sobolev (2000)

Journées équations aux dérivées partielles

We consider the operator in ℝ d , d ≥ 2 , of the form H = ( - Δ ) l + V , l > 0 with a function V periodic with respect to a lattice in ℝ d . We prove that the number of gaps in the spectrum of H is finite if 8 l > d + 3 . Previously the finiteness of the number of gaps was known for 4 l > d + 1 . Various approaches to this problem are discussed.

On the eigenvalues of a Robin problem with a large parameter

Alexey Filinovskiy (2014)

Mathematica Bohemica

We consider the Robin eigenvalue problem Δ u + λ u = 0 in Ω , ∂ u / ∂ ν + α u = 0 on ∂ Ω where Ω ⊂ ℝ n , n ≥ 2 is a bounded domain and α is a real parameter. We investigate the behavior of the eigenvalues λ k ( α ) of this problem as functions of the parameter α . We analyze the monotonicity and convexity properties of the eigenvalues and give a variational proof of the formula for the derivative λ 1 ' ( α ) . Assuming that the boundary ∂ Ω is of class C 2 we obtain estimates to the difference λ k D - λ k ( α ) between the k -th eigenvalue of the Laplace operator with Dirichlet...

On the existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems.

Jacqueline Fleckinger, Jesús Hernández, François De Thélin (2003)

RACSAM

We study the existence of principal eigenvalues for differential operators of second order which are not necessarily in divergence form. We obtain results concerning multiplicity of principal eigenvalues in both the variational and the general case. Our approach uses systematically the Krein-Rutman theorem and fixed point arguments for the inverse of the spectral radius of some associated problems. We also use a variational characterization for both the self-adjoint and the general case.

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