We provide an upper estimate for the eigenvalues of the curl curl operator on a bounded, three-dimensional Euclidean domain in terms of eigenvalues of the Dirichlet Laplacian. The result complements recent inequalities between curl curl and Neumann Laplacian eigenvalues. The curl curl eigenvalues considered here correspond to the Maxwell eigenvalue problem with constant material parameters.
We prove that the Laplace spectrum of the generic ellipse is simple, both with Neumann and Dirichlet boundary condition. We rely on the known multiplicities in the spectrum of the disk (Bourget's hypothesis) and on a refined version of our method of asymptotic separation of variables. In v1, the statement of prop. 2.1 is correct but the proof is not.
The description of all correct restrictions of the maximal operator are considered in a Hilbert space. A class of correct restrictions are obtained for which a similar transformation has the domain of the fixed correct restriction. The resulting theorem is applied to the study of n-order differentiation operator with singular coefficients.
In this paper we prove an asymptotic behavior for the radial eigenvalues to the Dirichlet $p$-Laplacian problem $-Δ_p\,u = λ\,|u|^{p-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is an annular domain $Ω=Ω_{R,\overline{R}}$ in $\mathbb{R}^N$.
We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results in particular extend some inequalities due to Kutller and Sigillito from subsets of $\mathbb{R}^2$ to the manifold setting.
The inverse nodal problem for Dirac type integro-differential operator with the spectral parameter in the boundary conditions is studied. We prove that dense subset of the nodal points determines the coefficients of differential part of operator and gives partial information for integral part of it.
A. Tajmouati, M. Amouch, M. R. F. Alhomidi Zakariya
In this paper, we give conditions for which the $C_0$ semigroups satisfies spectral equality for semiregular, essentially semiregular and semi-Fredholm spectrum. Also, we establish the spectral inclusion for B-Fredholm spectrum of a $C_0$ semigroups.
We study torsional rigidity for graph and quantum graph analogs of well-known pairs of isospectral non-isometric planar domains. We prove that such isospectral pairs are distinguished by torsional rigidity.
This paper is concerned with the study of theexistence/non-existence of the discrete spectrum of the Laplaceoperator on a domain of $\mathbb R ^3$ which consists in atwisted tube. This operator is defined by means of mixed boundaryconditions. Here we impose Neumann Boundary conditions on abounded open subset of the boundary of the domain (the Neumannwindow) and Dirichlet boundary conditions elsewhere.
We consider the elasticity operator with zero Poisson's ratio on an infinite strip and an infinite plate with a horizontal crack. We prove an asymptotic formula for the distance of the embedded eigenvalues to some spectral threshold of the operator as the crack becomes small.
In this paper, we study the Selberg and Ruelle zeta functions on compact hyperbolic odd dimensional manifolds. These zeta functions are defined on one complex variable $s$ in some right half-plane of $\mathbb{C}$. We use the Selberg trace formula for arbitrary not neccesarily unitary representations of the fundamental group to establish the meromorphic continuation of these zeta functions to the whole complex plane.
We consider the minimal differential operator A generated in $L^2(0,\infty)$ by the differential expression $l(y) = (-1)^n y^{(2n)}$. Using the technique of boundary triplets and the corresponding Weyl functions, we find explicit form of the characteristic matrix and the corresponding spectral function for the Friedrichs and Krein extensions of the operator A.
We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijel's original approach and an estimate due to Toth and Zelditch for the number of boundary zeros of Neumann eigenfunctions.
Inequalities are derived for sums and quotients of eigenvalues of magnetic Schroedinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.
We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.
We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation.
Let ${\cal D}$ denote the class of bounded real analytic plane domains with the symmetry of an ellipse. We prove that if $Ω_1, Ω_2 \in {\cal D}$ and if the Dirichlet spectra coincide, $Spec(Ω_1) = Spec(Ω_2)$, then $Ω_1 = Ω_2$ up to rigid motion.