Spectral theory for perturbed Krein Laplacians in nonsmooth domains (original) (raw)
The essential spectrum of Neumann Laplacians on some bounded singular domains
Journal of Functional Analysis, 1991
In the present paper we consider Neumann Laplacians on singular domains of the type \rooms and passages" or \combs" and we show that, in typical situations, the essential spectrum can be determined from the geometric data. Moreover, given an arbitrary closed subset S of the non-negative reals, we construct domains = (S) such that the essential spectrum of the Neumann Laplacians on is just this set S.
A generic property for the eigenfunctions of the Laplacian
Topological Methods in Nonlinear Analysis, 2002
In this work we show that, generically in the set of C 2 bounded regions of R n , n ≥ 2, the inequality R Ω φ 3 = 0 holds for any eigenfunction of the Laplacian with either Dirichlet or Neumann boundary conditions.
Towards a reversed Faber–Krahn inequality for the truncated Laplacian
Revista Matemática Iberoamericana
We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for a class of very degenerate elliptic operators, with the aim to show that, at least for square type domains having fixed volume, the symmetry of the domain maximize the principal eigenvalue, contrary to what happens for the Laplacian.
Contemporary Mathematics, 2005
Let Ω ⊂ R d be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator -1 2 ∆ in Ω -Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ 0 (Ω, A) > 0 denote its principal eigenvalue. We discuss the question of monotonicity of λ 0 (Ω, A) in its dependence on the domain Ω. The main point of this note is to suggest an open problem that is in the spirit of Chavel's question concerning domain monotonicity for the Neumann heat kernal. Let Ω ⊂ R d be a bounded domain with smooth boundary and let A ⊂⊂ Ω be a smooth, compactly embedded subdomain. Consider the operator -1 2 ∆ in Ω -Ā with the Dirichlet boundary condition at ∂A and the Neumann boundary condition at ∂Ω, and let λ 0 (Ω, A) > 0 denote its principal eigenvalue. If instead of the Neumann boundary condition, one imposes the Dirichlet boundary condition at ∂Ω, then it's easy to see that λ 0 (Ω, A) is monotone decreasing in Ω and increasing in A. Similarly, in the case at hand, it is clear that λ 0 (Ω, A) is monotone increasing in A; however, the question of monotonicity in Ω is not easily resolved. The impetus for studying this question arose in part from a recent paper in which one can find the asymptotic behavior of λ 0 (Ω, A) when A is a ball that shrinks to a point,
Spectral Properties of the Dirichlet-To-Neumann Operator on Lipschitz Domains
2007
The Dirichlet-to-Neumann operator D! is defined on L 2 (!) where ! is the boundary of a Lipschitz domain " and ! a real number which is not an eigenvalue of the Dirichlet Laplacian on L2("). We show that D! is a selfadjoint lower bounded operator with compact resolvent. There is a close connection between its eigenvalues and those of the Laplacian # µ on L 2 (") with Robin boundary conditions "u = µu|! where µ ! R. This connection is used to generalize L. Friedlander's result ! N+1 " ! D ,k =1 ,2 (where ! D is the k # th Dirichlet and ! N the k # th Neumann eigenvalue) to Lipschitz domains. We show that this Euclidean result is false, though, if an arbitrary compact Riemannian manifold M is considered instead of R d and " is suitable domain in M.
The spectral estimates for the Neumann–Laplace operator in space domains
Advances in Mathematics
In this paper we prove discreteness of the spectrum of the Neumann-Laplacian (the free membrane problem) in a large class of non-convex space domains. The lower estimates of the first non-trivial eigenvalue are obtained in terms of geometric characteristics of Sobolev mappings. The suggested approach is based on Sobolev-Poincaré inequalities that are obtained with the help of a geometric theory of composition operators on Sobolev spaces. These composition operators are induced by generalizations of conformal mappings that is called as mappings of bounded 2-distortion (weak 2-quasiconformal mappings).
The Laplacian with Robin boundary conditions on arbitrary domains
2003
Using a capacity approach, we prove in this article that it is always possible to define a realization µ of the Laplacian on L 2 ( ) with generalized Robin boundary conditions where is an arbitrary open subset of R n and µ is a Borel measure on the boundary ∂ of . This operator µ generates a sub-Markovian C 0 -semigroup on L 2 ( ). If dµ = β dσ where β is a strictly positive bounded Borel measurable function defined on the boundary ∂ and σ the (n−1)-dimensional Hausdorff measure on ∂ , we show that the semigroup generated by the Laplacian with Robin boundary conditions β has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L p ( ) is independent of p ∈ [1, ∞). Our approach constitutes an alternative way to Daners who considers the (n − 1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture. : 31C15, 35J25, 46E35, 47D03, 47D07.
Journal d'Analyse Mathématique, 2011
This paper has two main goals. First, we are concerned with a description of all self-adjoint extensions of the Laplacian − C ∞ 0 ( ) in L 2 ( ; d n x). Here, the domain belongs to a subclass of bounded Lipschitz domains (which we term quasi-convex domains), that contains all convex domains as well as all domains of class C 1,r , for r > 1/2. Second, we establish Kreȋn-type formulas for the resolvents of the various self-adjoint extensions of the Laplacian in quasiconvex domains and study the well-posedness of boundary value problems for the Laplacian as well as basic properties of the corresponding Weyl-Titchmarsh operators (or energy-dependent Dirichlet-to-Neumann maps). One significant innovation in this paper is an extension of the classical boundary trace theory for functions in spaces that lack Sobolev regularity in a traditional sense, but are suitably adapted to the Laplacian.