On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

The treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ?_λ to the equation ?_λ – K λ ? λ = fλ as λ → 0 where {K_λ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I - K_λ (I identity) is bijective for λ∈(0,α) whilst I - K₀ is not and ‖ K_λ – K₀‖ →, 0, ‖f_λ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non-connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.     

On turning points

For a family {T + N_λ: λ ? [a, b]} of semilinear operators T + N_λ in L₂(Ω) the solution set {(λ, u_λ) ? J × D(T): Tu_λ + N_λu_λ = h} is investigated with respect to turning points. By Ljapunov-Schmidt-reduction and calculation of the derivatives of the bifurcation equation a class of turning points is characterized by properties of these derivatives.     

On the Ratio of Consecutive Eigenvalues in N-Dimensions

It is proved that λ₂ ≤ (1 + 4/N)λ₁ where λ₁ ≤ λ₂ ≤ ··· are the eigenvalues of the N-dimensional Laplace operator in a domain D with vanishing boundary conditions. This extends the N = 2 case of Payne, Pólya and Weinberger and the result is independent of the domain D.     

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on Z ^N with symbols which are bounded on Z ^N ×T ^N together with their derivatives with respect to the second variable. In the same way as partial differential operators on R ^N are included in an algebra of pseudodifferential operators, difference operators on Z ^N are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces l _w ^p (Z ^N ) and to Phragmen–Lindelöf type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrödinger operators and the decay of their eigenfunctions at infinity.     

Multipliers of Laplace transform type for ultraspherical expansions

We define and investigate the multipliers of Laplace transform type associated to the differential operator L_λf (θ) = –f ″(θ) – 2λ cot θf ′(θ) + λ²f (θ), λ > 0. We prove that these operators are bounded in L^p ((0, π), dm_λ) and of weak type (1, 1) with respect to the same measure space, dm_λ (θ) = (sin θ)^2λ dθ. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Fast Computations with the Inverse to Harmonic Potential Operators via Domain Decomposition

In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operations in terms of the Poincare-Steklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can be efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs are of the order O(N log^p N) and O(N log² N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations.     

An Inverse Function Theorem for Fréchet Spaces Satisfying a Smoothing Property and (DN)

Classical inverse function theorems of Nash-Moser type are proved for Fréchet spaces that admit smoothing operators as introduced by Nash. In this note an inverse function theorem is proved for Fréchet spaces which only have to satisfy the condition (DN) of Vogt and the smoothing property (S_Ω)_t; for instance, any Fréchet-Hilbert space which is an (Ω)-space in standard form has property (S_Ω)_t. The main result of this paper generalizes a theorem of Lojasiewicz and Zehnder. It can be applied to the space C^∞(K) if the compact K ? ?^N is the closure of its interior and subanalytic; different from classical results the boundary of K may have singularities like cusps. The growth assumptions on the mappings are formulated in terms of the weighted multiseminorms [ ]m,k introduced in this paper; nonlinear smooth partial differential operators on C^∞(K) and their derivatives satisfy these formal assumptions.     

Multiplicative perturbations of the Laplacian and related approximation problems

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on \mathbbR^N,  N 3 1{{\mathbb{R}}^{N},\; N\geq1} . It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C ₀-semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on \mathbbR^N{{\mathbb{R}}^{N}} . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.     

Fast computations with the harmonic Poincaré–Steklov operators on nested refined meshes

In this paper we develop asymptotically optimal algorithms for fast computations with the discrete harmonic Poincaré–Steklov operators (Dirichlet–Neumann mapping) for interior and exterior problems in the presence of a nested mesh refinement. Our approach is based on the multilevel interface solver applied to the Schur complement reduction onto the nested refined interface associated with a nonmatching decomposition of a polygon by rectangular substructures. This paper extends methods from Khoromskij and Prössdorf (1995), where the finite element approximations of interior problems on quasi‐uniform grids have been considered. For both interior and exterior problems, the matrix–vector multiplication with the compressed Schur complement matrix on the interface is shown to have a complexity of the order O(N _r log³ N _u), where N_r = O((1 + p _r) N _u) is the number of degrees of freedom on the polygonal boundary under consideration, N _u is the boundary dimension of a finest quasi‐uniform level and p _r ⩾ 0 defines the refinement depth. The corresponding memory needs are estimated by O(N _r log^q N _u), where q = 2 or q = 3 in the case of interior and exterior problems, respectively.

     

Fortran codes for computing the discrete Helmholtz integral operators

In this paper Fortran subroutines for the evaluation of the discrete form of the Helmholtz integral operators L _k, M _k, M _k ^t and N _k for two-dimensional, three-dimensional and three-dimensional axisymmetric problems are described. The subroutines are useful in the solution of Helmholtz problems via boundary element and related methods. The subroutines have been designed to be easy to use, reliable and efficient. The subroutines are also flexible in that the quadrature rule is defined as a parameter and the library functions (such as the Hankel, exponential and square root functions) are called from external routines. The subroutines are demonstrated on test problems arising from the solution of the Neumann problem exterior to a closed boundary via the Burton and Miller equation. This revised version was published online in August 2006 with corrections to the Cover Date.     

Constructive existence theorems for problems of Thomas-Fermi Type

A minimal positive solution of the Thomas-Fermi problem ? = λt^?1/2 w^3/2, w(0) = 1, w(1) = w(1) is shown to exist for each λ > 0. It is proved that all positive solutions, for a given value of λ, are strictly ordered and that the minimal positive solution w_λ is a decreasing function of λ. Upper and lower analytic bounds for w _λ are given and these bounds are shown to initiate sequences of Picard and Newton iterates which converge monotonically to w _λ. A comparative analysis of the efficiency of the iteration schemes is presented. The methods used are of a general nature and can be applied to a variety of nonlinear boundary value problems of convex type [14].     

On the Logarithm Component in Trace Defect Formulas

In asymptotic expansions of resolvent traces Tr(A(P ? λ)^?1) for classical pseudodifferential operators on closed manifolds, the coefficient C ₀(A, P) of ( ? λ)^?1 is of special interest, since it is the first coefficient containing nonlocal elements from A; moreover, it enters in index formulas. C ₀(A, P) also equals the zeta function value at zero when P is invertible. C ₀(A, P) is a trace modulo local terms, since C ₀(A, P) ? C ₀(A, P′) and C ₀([A, A′], P) are local. By use of complex powers P ^s (or similar holomorphic families of order s), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from A, A′, log P and log P′.

The present paper has two purposes. One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of P as in the original proofs. We also give here a simple direct proof of a recent residue formula of Scott for C ₀(I, P). The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.     

Weyl-Titchmarsh functions of vector-valued Sturm-Liouville operators on the unit interval

The matrix-valued Weyl-Titchmarsh functions M(λ) of vector-valued Sturm-Liouville operators on the unit interval with the Dirichlet boundary conditions are considered. The collection of the eigenvalues (i.e., poles of M(λ)) and the residues of M(λ) is called the spectral data of the operator. The complete characterization of spectral data (or, equivalently, N×N Weyl-Titchmarsh functions) corresponding to N×N self-adjoint square-integrable matrix-valued potentials is given, if all N eigenvalues of the averaged potential are distinct.     

Trace formulas for a class of subnormal tuples of operators

Trace formulas are established for the product of commutators related to subnormal tuple of operators (S ₁,...,S _n ) with minimal normal extension (N ₁,...,N _n ) satisfying conditions that sp(S _j )/sp(N _j ) is simply-connected with smooth boundary Jordan curve sp(N _i ) and [S _j ^* ,S _j ]^1/2 L ¹,j=1, 2,...,n.Some complete unitary invariants related to the trace formulas are found.This work is supported in part by NSF Grant no. DMS-9101268.     

Regularity for Shape Optimizers: The Degenerate Case

We consider minimizers of (1) where F is a function nondecreasing in each parameter, and λ_k(Ω) is the k^th Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often-studied F=λ_N. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler-Lagrange equation for the minimizers that is not otherwise available. © 2019 Wiley Periodicals, Inc.     

Compact differences of composition operators on Holomorphic Function Spaces in the unit ball

We find a lower bound for the essential norm of the difference of two composition operators acting on H 2(BN ) or As2 (BN ) (s > 1). This result plays an important role in proving a necessary and sufficient condition for the difference of linear fractional composition operators to be compact, which answers a question posed by MacCluer and Weir in 2005.     

Über das wachstum von resolventen in der nähe einer isolierten singularität

Let L(λ) be a bundle of linear bounded operators between two Banach spaces. In this paper we study the behaviour of {L(λ)}^?1, if λ tends to λ_o and L(λ_o) is a Fredholm operator with index 0. We show that the growth of this resolvent can be described by the length of certain chains of generalized principal vectors; if L(λ) depends analytically on the parameter λ, we get a complete characterization for an isolated singularity of L, and also a Laurent expansion for the resolvent. Finally, we give applications to a broad class of bundles of bounded self-adjoint operators.     

On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region G ⊂ ℝ² are considered. The domain of these operators consists of functions in the Sobolev space W ₂ ^m (G) that are generalized solutions of the corresponding elliptic equation of order 2m with the right-hand side in L ₂(G) and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 116–135.     

Alternating Sign Multibump Solutions of Nonlinear Elliptic Equations in Expanding Tubular Domains

Let Γ denote a smooth simple curve in ? ^N , N ≥ 2, possibly with boundary. Let Ω _R be the open normal tubular neighborhood of radius 1 of the expanded curve RΓ: = {Rx | x ∈ Γ??Γ}. Consider the superlinear problem ? Δu + λu = f(u) on the domains Ω _R , as R → ∞, with homogeneous Dirichlet boundary condition. We prove the existence of multibump solutions with bumps lined up along RΓ with alternating signs. The function f is superlinear at 0 and at ∞, but it is not assumed to be odd. If the boundary of the curve is nonempty our results give examples of contractible domains in which the problem has multiple sign changing solutions.