On a class of scalar variational inequalities with measure data

We consider a class of variational inequalities defining the so-called hysteresis play operator. We propose a new approach to discontinuous BV-solutions based on measure theoretical arguments, which enable us to infer the existence of solutions as a simple consequence of the classical theory. In this way, we generalize a recent result where only continuous BV-solutions were studied. We also provide a representation formula which allows to deduce the continuity of the play operator from general theorems on hysteresis operators.     

BV ‐extension of rate independent operators

Rate independent operators naturally arise in the mathematical analysis of hysteresis. Among rate independent operators, the locally monotone ones are those better suited for the study of PDE's with hysteresis. We prove that a rate independent operator R: Lip (0, T) → BV (0, T) ∩ C (0, T) which is locally monotone and continuous with respect to the strict topology of BV admits a unique continuous extension R?: BV (0, T) → BV (0, T). This general result applies to several concrete hysteresis operators. For many of these operators the existence of a continuous extension was previously known at most to the space BV (0, T) ∩ C (0, T). (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hankel-type operators on the space of bounded harmonic functions

We study Hankel-type operators on the space of bounded harmonic functions on the open unit disk. These operators are related to tight uniform algebras, the Dunford-Pettis property, and Bourgain algebras.

     

The play operator,the truncated variation and the generalisation of the Jordan decomposition

The play operator minimalizes the total variation on intervals, [0,T],T > 0, of functions approximating uniformly given regulated function with given accuracy and starting from a given point. In this article, we link the play operator with the so‐called truncated variation functionals, introduced recently by the second‐named author, and provide a semi‐explicit expression for the play operator in terms of these functionals. Generalisation for time‐dependent boundaries is also considered. This gives the best possible lower bounds for the total variation of the outputs of the play operator and its Jordan‐like decomposition. Copyright © 2014 John Wiley & Sons, Ltd.     

Compactness of the solution operator for a linear evolution equation with distributed measures

The main goal of the present paper is to define the solution operator associated to the evolution equation , , where generates a -semigroup in a Banach space , , , and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to BARAS, HASSAN, VERON, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.     

The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable subsets E of Ω. Here f_h⁺, f^? denote quadratic potentials defined on Ω¯×{symmetric d×d matrices}, h is the minimum energy of f_h⁺ and ε(u) is the symmetric gradient of the displacement field u. An equilibrium state û, Ê of J(u,E) is called one‐phase if E=?? or E=Ω, two‐phase otherwise. For two‐phase states, σ∣?E∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h??, σ>0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J(u,E) in the limit σ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.     

Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators

In this paper first we describe all (not necessarily linear or bijective) transformations on R^d${\mathbb{R}}^d$ with 2≤d<∞$2\le d<\infty$ which preserve the area of parallelograms spanned by any two vectors. We also characterize those (not necessarily linear) bijections on an arbitrary real Hilbert space that preserve the latter quantity. This answers a question raised by Rassias and Wagner, and it can be considered as a variant of the famous Wigner theorem on real Hilbert spaces which plays an important role in quantum mechanics. As a consequence, we solve a preserver problem of Molnár and Timmermann which has remained open stubbornly only in the two-dimensional case. Finally this two-dimensional result will be applied in order to strengthen their theorem in higher dimensions.     

Computable upper bounds for the constants in Poincaré‐type inequalities for fields of bounded deformation

We collect various Poincaré‐type inequalities valid for fields of bounded deformation and give explicit upper bounds for the constants being involved. Copyright © 2011 John Wiley & Sons, Ltd.     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

A Note on the Stability of the Integral-Differential Equation of the Hyperbolic Type in a Hilbert Space

In this paper, the initial-value problem for integral-differential equation of the hyperbolic type in a Hilbert space H is considered. The unique solvability of this problem is established. The stability estimates for the solution of this problem are obtained. The difference scheme approximately solving this problem is presented. The stability estimates for the solution of this difference scheme are obtained. In applications, the stability estimates for the solutions of the nonlocal boundary problem for one-dimensional integral-differential equation of the hyperbolic type with two dependent limits and of the local boundary problem for multidimensional integral-differential equation of the hyperbolic type with two dependent limits are obtained. The difference schemes for solving these two problems are presented. The stability estimates for the solutions of these difference schemes are obtained.     

Estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space

We obtain an estimate of the norm of the Lagrange interpolation operator in a multidimensional Sobolev space. It is shown that, under a suitable choice of the sequence of multi-indices, interpolation polynomials converge to the interpolated function and their rate of convergence is of the order of the best approximation of this function.     

The equivalence relations between the best approximation and linear operators approximation in a Besov space

In this paper, we characterize Besov type space introduced by the best approximation, best approximating elements and a kind of linear operators. The project is supported by NSFC     

The space for nondoubling measures in terms of a grand maximal operator

Let be a Radon measure on , which may be nondoubling. The only condition that must satisfy is the size condition , for some fixed . Recently, some spaces of type and were introduced by the author. These new spaces have properties similar to those of the classical spaces and defined for doubling measures, and they have proved to be useful for studying the boundedness of Calderón-Zygmund operators without assuming doubling conditions. In this paper a characterization of the new atomic Hardy space in terms of a maximal operator is given. It is shown that belongs to if and only if , and , as in the usual doubling situation.

     

The best quadrature formula based on Hermite information for the class KW~2[a,b]

The best quadrature formula has been found in the following sense:for afunction whose norm of the second derivative is bounded by a given constant and thebest quadrature formula for the approximate evaluation of integration of that function canminimize the worst possible error if the values of the function and its derivative at certainnodes are known.The best interpolation formula used to get the quadrature formula aboveis also found.Moreover,we compare the best quadrature formula with the open compoundcorrected trapezoidal formula by theoretical analysis and stochastic experiments.     

An example of the geometry of a 5th-order ODE: The metric on the space of conics in CP2

As an application of the method of [4], we find the metric and connection on the space of conics in ${\mathbb{CP}}^2$ determined as the solution space of the ODE (1). These calculations underpin the twistor construction of the Radon transform on conics in ${\mathbb{CP}}^2$ described in [5]. Two further examples of the method are provided.     

The best quadrature formula based on Hermite information for the class <Emphasis Type="Italic">KW</Emphasis><Superscript>2</Superscript> [<Emphasis Type="Italic">a,b</Emphasis>]

The best quadrature formula has been found in the following sense: for a function whose norm of the second derivative is bounded by a given constant and the best quadrature formula for the approximate evaluation of integration of that function can minimize the worst possible error if the values of the function and its derivative at certain nodes are known. The best interpolation formula used to get the quadrature formula above is also found. Moreover, we compare the best quadrature formula with the open compound corrected trapezoidal formula by theoretical analysis and stochastic experiments.