SVEP and compact perturbations

Let H be a complex separable infinite dimensional Hilbert space. In this paper, we prove that an operator T acting on H is a norm limit of those operators with single-valued extension property (SVEP for short) if and only if T^?, the adjoint of T, is quasitriangular. Moreover, if T^? is quasitriangular, then, given an ε>0, there exists a compact operator K on H with ‖K‖<ε such that T+K has SVEP. Also, we investigate the stability of SVEP under (small) compact perturbations. We characterize those operators for which SVEP is stable under (small) compact perturbations.     

A new approach to numerical solution of fixed-point problems and its application to delay differential equations

In this paper we consider a certain approximation of fixed-points of a continuous operator A mapping the metric space into itself by means of finite dimensional ε(h)-fixed-points of A. These finite dimensional functions are obtained from functions defined on discrete space grid points (related to a parameter h→0) by applying suitably chosen extension operators p_h. A theorem specifying necessary and sufficient conditions for existence of fixed-points of A in terms of ε(h)-fixed-points of A is given. A corollary which follows the theorem yields an approximate method for a fixed-point problem and determines conditions for its convergence. An example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided.Numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples.     

Nichtlineare Eigenwertprobleme mit zwei Parametern

In this paper we are searching for solutions (x,μdepending on (ε,τ)-of the nonlinear eigenvalue problem (A?λ)x+B(ε,x)+C(τ,x)=μx. Thus we are interested mainly in imposing such conditions on the analytic operators B(ε,x) and C(τ,x) that will ensure us to obtain solutions. Also we will discuss how these solutions depend onε andτ nearε=τ=0.     

Harmonic approximation of difference operators

For a general class of difference operators Hε=Tε+Vε on ?²(d(εZ)), where Vε is a multi-well potential and ε is a small parameter, we analyze the asymptotic behavior as ε→0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first n eigenvalues of Hε converge to the first n eigenvalues of the direct sum of harmonic oscillators on Rd located at the several wells. Our proof is microlocal.     

Multilinear extrapolation and applications to the bilinear Hilbert transform

We present two extrapolation methods for multi-sublinear operators that allow us to derive estimates for general functions from the corresponding estimates on characteristic functions. Of these methods, the first is applicable to general multi-sublinear operators while the second requires working with the so-called (ε,δ)-atomic operators. Among the applications, we discuss some new endpoint estimates for the bilinear Hilbert transform.     

On the convergence of bloch eigenfunctions in homogenization problems

We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with ε-periodic coefficients, where ε is a small parameter. Two cases are considered, the case of classical homogenization, where the coefficient matrix satisfies the ellipticity condition uniformly with respect to ε, and the case of two-scale homogenization, where the coefficient matrix has two phases and is highly contrast with hard-to-soft-phase contrast ratio 1: ε².     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.     

Multidimensional integral operators with bihomogeneous kernels: A projection method and pseudospectra

We consider multidimensional integral operators with bihomogeneous and rotation invariant kernels. For such operators we obtain a criterion for applicability of a projection method in the scalar and matrix cases and describe the limit behavior of the ε-pseudospectra of the truncated operators $A_{\tau _1 , \tau _2 } $ as τ ₁ → 0 and τ ₂ → 0.     

Conditional expectations on Riesz spaces

Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f-algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L¹.     

Convergence on layer-adapted meshes and anisotropic interpolation error estimates of non-standard higher order finite elements

For a general class of finite element spaces based on local polynomial spaces E with Pp⊂E⊂Qp we construct a vertex-edge-cell and point-value oriented interpolation operators that fulfil anisotropic interpolation error estimates.Using these estimates we prove ε-uniform convergence of order p for the Galerkin FEM and the LPSFEM for a singularly perturbed convection-diffusion problem with characteristic boundary layers.     

Boundedness of para-product operators on spaces of homogeneous type

We obtain the boundedness of Calderón-Zygmund singular integral operators T of non-convolution type on Hardy spaces H ^p (X) for 1/(1 + ε) < p ? 1, where X is a space of homogeneous type in the sense of Coifman and Weiss (1971), and ε is the regularity exponent of the kernel of the singular integral operator T. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was used in the literature.     

Extension operators via semigroups

The Roper-Suffridge extension operator and its modifications are powerful tools to construct biholomorphic mappings with special geometric properties. The first purpose of this paper is to analyze common properties of different extension operators and to define an extension operator for biholomorphic mappings on the open unit ball of an arbitrary complex Banach space. The second purpose is to study extension operators for starlike, spirallike and convex in one direction mappings. In particular, we show that the extension of each spirallike mapping is A-spirallike for a variety of linear operators A. Our approach is based on a connection of special classes of biholomorphic mappings defined on the open unit ball of a complex Banach space with semigroups acting on this ball.     

Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients

We derive global Carleman estimates for one-dimensional linear parabolic equations t_∂±x_∂(cx_∂) with a coefficient of bounded variations. These estimates are obtained by approximating c by piecewise constant coefficients, cε, and passing to the limit in the Carleman estimates associated to the operators defined with cε. Such estimates yields observability inequalities for the considered linear parabolic equation, which, in turn, yield controllability results for classes of semilinear equations.     

Direct sums and the Szlenk index

For α an ordinal and 1<p<∞, we determine a necessary and sufficient condition for an ?p-direct sum of operators to have Szlenk index not exceeding ωα. It follows from our results that the Szlenk index of an ?p-direct sum of operators is determined in a natural way by the behaviour of the ε-Szlenk indices of its summands. Our methods give similar results for c₀-direct sums.     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).     

The extension of functions of Sobolev classes beyond the boundary of the domain on Carnot groups

We prove the theorem on extension of the functions of the Sobolev space W _p ^l (Ω) which are defined on a bounded (ε, δ)-domain Ω in a two-step Carnot group beyond the boundary of the domain of definition. This theorem generalizes the well-known extension theorem by P. Jones for domains of the Euclidean space.     

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever $T(x)\cap S(x)\not=\emptyset $ for every x?∈?D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their ε-subdifferentials intersect at every point of that domain.     

Fully nonlinear singularly perturbed equations and asymptotic free boundaries

In this paper we study one-phase fully nonlinear singularly perturbed elliptic problems with high energy activation potentials, ζε(u) with ζε→δ₀⋅∫ζ. We establish uniform and optimal gradient estimates of solutions and prove that minimal solutions are non-degenerated. For problems governed by concave equations, we establish uniform weak geometric properties of approximating level surfaces. We also provide a thorough analysis of the free boundary problem obtained as a limit as the ε-parameter term goes to zero. We find the precise jumping condition of limiting solutions through the phase transition, which involves a subtle homogenization process of the governing fully nonlinear operator. In particular, for rotational invariant operators, F(D²u), we show the normal derivative of limiting function is constant along the interface. Smoothness properties of the free boundary are also addressed.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">C</Emphasis>)-Isometric Operators

In this paper we study the spectral properties of (m, C)-isometric operators. In particular, if \(T\in \mathcal{{L(H)}}\) is (m, C)-isometric operators, then the power of (m, C)-isometric operators is also (m, C)-isometric operators. Moreover, if \(T^{*}\) has the single-valued extension property, then T has the single-valued extension property. Finally, we investigate conditions for (m, C)-isometric operators to be (1, C)-isometric operators.