The inclusion of Stekloff eigenvalues by boundary data approximation

We give eigenvalue inclusion theorems allowing the computation of both-sided Stekloff-eigenvalue bounds from a given approximate eigenpair. We demonstrate in numerical examples the possibility to get close bounds for even higher eigenvalues using the boundary collocation method. A good choice for trial functions are singularity functions allowing a control for the numerical stability and—in theory—an eigenvalue inclusion in an arbitrary small interval.The inclusion theorems may be interpreted as a priori norm inequalities, too.

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Zusammenfassung Wir stellen Eigenwerteinschlußsätze vor, welche die Berechnung beiderseitiger Eigenwertschranken in Abhängigkeit von vorgelegten Näherungseigenpaaren gestatten. Wir demonstrieren an numerischen Beispielen, daß es mit dem Kollokationsverfahren möglich ist, sehr gute Einschlüsse selbst höherer Eigenwerte zu erhalten. Als Ansatzfunktionen bewähren sich Singularitätenfunktionen, welche eine Steuerbarkeit der numerischen Stabilität ermöglichen. Zudem erlauben diese Funktionen in der Theorie einen Einschluß in beliebiger Güte.Die Einschlußsätze können auch als a priori Norm-Ungleichungen angesehen werden.

     

Sums of reciprocal Stekloff eigenvalues

Let 0 = λ₁ < λ₂ ≤ λ₃ ≤ … be the Stekloff eigenvalues of a plane domain. The paper is concerned with formulas for ∑^∞₂λ^(–2)_j in simply and doubly connected domains. In the simply connected case it is proven that the disk minimized this sum. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,,) be an n(2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems

where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first non-zero eigenvalues of the above problems will be denoted by p₁ and q₁, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius , here λ₁ denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q₁nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius . Finally, we show that q₁A/V and that if the equality holds and if there is a point x₀∂M such that the mean curvature of ∂M at x₀ is no less than A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.     

Inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains of

We prove universal inequalities for eigenvalues of the buckling problem of arbitrary order on bounded domains in which are independent of the domains.     

Equilibria for set-valued maps on nonsmooth domains

A set-valued map defined on a compact lipschitzian retract of a normed space with nontrivial Euler characteristic and satisfying (i) a strong graph approximation property and (ii) a tangency condition expressed in terms of Clarke’s tangent cone, admits an equilibrium. This result extends in a simple way known solvability theorems to a large class of nonconvex set-valued maps defined on nonsmooth domains. Dedicated to Professor Felix Browder     

On the Hadamard formula for nonsmooth domains

We consider the first eigenvalue of the Dirichlet-Laplacian in three cases: C1,1-domains, Lipschitz domains, and bounded domains without any smoothness assumptions. Asymptotic formula for this eigenvalue is derived when domain subject arbitrary perturbations. For Lipschitz and arbitrary nonsmooth domains, the leading term in the asymptotic representation distinguishes from that in the Hardamard formula valid for smooth perturbations of smooth domains. For asymptotic analysis we propose and prove an abstract theorem demonstrating how eigenvalues vary under perturbations of both operator in Hilbert space and Hilbert space itself. This abstract theorem is of independent interest and has substantially broader field of applications.     

Elliptic boundary-value problems in nonsmooth domains

In a bounded domain, we study elliptic boundary-value problems for equations and systems of the Douglis-Nirenberg structure in complete scales of Banach spaces. The boundary of the domain contains conic points, edges, etc. A theorem on local increase in the smoothness of generalized solutions and a theorem on complete collection of isomorphisms are proved. Applications are considered. It is shown that the results obtained are also valid for transmission problems, nonlocal elliptic problems, elliptic problems with a parameter, and parabolic problems.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 701–709, May, 1995.This research was partially supported by a grant of the American Mathematical Society and by the Ukrainian State Committee on Science and Technology.     

An inequality for Steklov eigenvalues for planar domains

Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Nonsymmetric systems on nonsmooth planar domains

We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.

     

Inclusion domains for the eigenvalues of stochastic matrices

Summary Inclusion domains for the nontrivial eigenvalues of stochastic matrices are given which are closely related to a bound for the nontrivial eigenvalues given by Bauer, Deutsch and Stoer. The inclusion domains are constructed by adapting Bauer's concept of a field of values subordinate to norms to the more general case of seminorms.Parts of this paper were presented at the Gatlinburg Symposium on Numerical Algebra in 1969.     

Singular solutions for semi-linear parabolic equations on nonsmooth domains

We prove the existence of positive singular solutions for the semi-linear parabolic equation on Ω=D×]0,∞[, where p>1,D is a bounded NTA-domain in Rn, n?2, and μ is in a general class of signed Radon measures on D covering the elliptic Kato class of potentials adopted by Zhang and Zhao. A new proof of the result based on a simple fixed point theorem is also given.     

The hp finite element method for singularly perturbed problems in nonsmooth domains

We consider the numerical approximation of singularly perturbed elliptic boundary value problems over nonsmooth domains. We use a decomposition of the solution that contains a smooth part, a corner layer part and a boundary layer part. Explicit guidelines for choosing mesh‐degree combinations are given that yield finite element spaces with robust approximation properties. In particular, we construct an hp finite element space that approximates all components uniformly, at a near exponential rate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 63–89, 1999     

Second-order conditions for nonsmooth multiobjective optimization problems with inclusion constraints

This paper investigates second-order optimality conditions for general multiobjective optimization problems with constraint set-valued mappings and an arbitrary constraint set in Banach spaces. Without differentiability nor convexity on the data and with a metric regularity assumption the second-order necessary conditions for weakly efficient solutions are given in the primal form. Under some additional assumptions and with the help of Robinson -Ursescu open mapping theorem we obtain dual second-order necessary optimality conditions in terms of Lagrange-Kuhn-Tucker multipliers. Also, the second-order sufficient conditions are established whenever the decision space is finite dimensional. To this aim, we use the second-order projective derivatives associated to the second-order projective tangent sets to the graphs introduced by Penot. From the results obtained in this paper, we deduce and extend, in the special case some known results in scalar optimization and improve substantially the few results known in vector case.     

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

Global higher integrability for parabolic quasiminimizers in nonsmooth domains

We study the global higher integrability of the gradient of a parabolic quasiminimizer with quadratic growth conditions. We show that if the lateral boundary satisfies a capacity density condition and if boundary and initial values are smooth enough, then quasiminimizers globally belong to a higher Sobolev space than assumed a priori. We derive estimates near the lateral and the initial boundaries.