Domain Dependence for Eigenvalues of the Neumann Problem for Elliptic Operators (original) (raw)
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This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. As an application, we develop a theory for the Laplacian in Lipschitz domains. In particular, if the domains are assumed to be C 1,α regular, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result.
2015
This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class C 1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case, however, in the C 1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C 1. Moreover, as an application we consider the case of a C 1-perturbation when the reference domain is of class C 1,α .
Analysis & PDE, 2015
This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class C 1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case however, in the C 1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C 1. Moreover, as an application we consider the case of a C 1-perturbation when the reference domain is of class C 1,α .
Domain dependence of eigenvalues of elliptic type operators
Mathematische Annalen, 2013
The dependence on the domain is studied for the Dirichlet eigenvalues of an elliptic operator considered in bounded domains. Their proximity is measured by a norm of the difference of two orthogonal projectors corresponding to the reference domain and the perturbed one; this allows to compare domains that have non-smooth boundaries and different topology. The main result is an asymptotic formula in which the remainder is evaluated in terms of this quantity. As an application, the stability of eigenvalues is estimated by virtue of integrals of squares of the gradients of eigenfunctions for elliptic problems in different domains. It occurs that these stability estimates imply well-known inequalities for perturbed eigenvalues.
Zeitschrift für Analysis und ihre Anwendungen, 2000
We calculate the main asymptotic terms for eigenvalues, both simple and multiple, and eigenfunctions of the Neumann Laplacian in a three-dimensional domain Ω(h) perturbed by a small (with diameter O(h)) Lipschitz cavern ω h in a smooth boundary ∂Ω = ∂Ω(0). The case of the hole ω h inside the domain but very close to the boundary ∂Ω is under consideration as well. It is proven that the main correction term in the asymptotics of eigenvalues does not depend on the curvature of ∂Ω while terms in the asymptotics of eigenfunctions do. The influence of the shape of the cavern to the eigenvalue asymptotics relies mainly upon a certain matrix integral characteristics like the tensor of virtual masses. Asymptotically exact estimates of the remainders are derived in weighted norms.
An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator
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The Neumann problem is in general "harder" than the Dirichlet problem. In this talk we show in certain cases they are "equally hard"/equivalent, in the sense that solving one of them leads to the solution of the other one. More precisely, we give a representation of the solution of the Neumann problem for the Laplace operator on the unit ball in R n (n ≥ 1) in terms of the solution of an associated Dirichlet problem. The representation is suitable for extensions, and we provide extensions to: a) other operators besides the Laplacian b) smooth planar domains c) infinite dimensional case d) general boundary data. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.
Domain Perturbation for the first Eigenvalue of the Dirichlet Schrödinger Operator
Partial Differential Operators and Mathematical Physics, 1995
Let ft C R n be an open connected set. We consider the Dirichlet-Schrodinger operator H-A^ + V on L 2 {Q) (where A^ denotes the Laplacian with Dirichlet boundary conditions and V is a suitable potential). In a recent paper, F. Gesztesy and Z. Zhao [15] showed that the first eigenvalue A(H) of H is a strictly monotonic function with respect to the domain ft (up to capacity, see below for the precise statement). Their proof is given with help of probabilistic methods. The purpose of this article is to give an analytic proof of this result. In fact, we prove a generalization, allowing the potential to vary as well. Our proof is based on a domination argument for positive irreducible semigroups (Section 2). In the main theorem (Theorem 3.1), the difference of two open sets is measured by capacity. Some results concerning this notion are established in Section 1. In particular, we give a short proof of the fact that using the characterization of closed order ideals in H 1^) which has been given recently by Stollmann [24]. This seems to be of independent interest.
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fy methods of stohsti nlysis on iemnnin mnifoldsD we derive expliit onstnts c 1 @DA nd c 2 @DA for dEdimensionl ompt iemnnin mnifold D with oundry suh tht c 1 @DA p kk 1 6 krk 1 6 c 2 @DA p kk 1 holds for ny hirihlet eigenfuntion of ¡ with eigenvlue F sn prtiulrD when D is onvex with nonEnegtive ii urvtureD the estimte holds for c 1 @DA a I de ; c 2 @DA a p e p P p C p R p P ! : gorresponding twoEsided grdient estimtes for xeumnn eigenfuntions re derived in the seE ond prt of the pperF
Hadamard asymptotics for eigenvalues of the Dirichlet Laplacian
arXiv: Analysis of PDEs, 2018
This paper is dedicated to the classical Hadamard formula for asymptotics of eigenvalues of the Dirichlet-Laplacian under perturbations of the boundary. We prove that the Hadamard formula still holds for C1C^1C1-domains with C1C^1C1-perturbations. We also derive an optimal estimate for the remainder term in the C1,alphaC^{1,\alpha}C1,alpha-case. Furthermore, if the boundary is merely Lipschitz, we show that the Hadamard formula is not valid.