A band limited and Besov class functional calculus for Tadmor-Ritt operators

For Banach space operators T satisfying the Tadmor-Ritt condition a band limited H^∞ calculus is established, where and a is at most of the order C(T)⁵. It follows that such a T allows a bounded Besov algebra B_{∞ 1}⁰ functional calculus, These estimates are sharp in a convenient sense. Relevant embedding theorems for B_{∞ 1}⁰ are derived. Received: 25 October 2004; revised: 31 January 2005     

The Weiss conjecture on admissibility of observation operators for contraction semigroups

We prove the conjecture of George Weiss for contraction semigroups on Hilbert spaces, giving a characterization of infinite-time admissible observation functionals for a contraction semigroup, namely that such a functionalC is infinite-time admissible if and only if there is anM>0 such that for alls in the open right half-plane. HereA denotes the infinitesimal generator of the semigroup. The result provides a simultaneous generalization of several celebrated results from the theory of Hardy spaces involving Carleson measures and Hankel operators.     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

Profile of Blow-up Solution to Hyperbolic System with Nonlocal Term

This paper is concerned with a nonlocal hyperbolic system as follows utt = △u ＋ （∫Ωvdx ）^p for x∈R^N,t〉0 ,utt = △u ＋ （∫Ωvdx ）^q for x∈R^N,t〉0 ,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N,u（x,0）=u0（x）,ut（x,0）=u01（x） for x∈R^N, where 1≤ N ≤3, p ≥1, q ≥ 1 and pq 〉 1. Here the initial values are compactly supported and Ω belong to R^N is a bounded open region. The blow-up curve, blow-up rate and profile of the solution are discussed.     

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Total Variation Distance for Poisson Subset Numbers

Let n be an integer and A₀,..., A_k random subsets of {1,..., n} of fixed sizes a₀,..., a_k, respectively chosen independently and uniformly. We provide an explicit and easily computable total variation bound between the distance from the random variable , the size of the intersection of the random sets, to a Poisson random variable Z with intensity λ = EW. In particular, the bound tends to zero when λ converges and for all j = 0,..., k, showing that W has an asymptotic Poisson distribution in this regime. Received February 24, 2005     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Multiple Boundary Concentrating Solutions to Dirichlet Problem of Hénon Equation

Abstract  Let Ω be the unit ball centered at the origin in . We study the following problem

Maximal dissipativity for degenerate Kolmogorov operators

In this paper we are interested in studying the properties of an elliptic degenerate operator N₀ in the space L^p of with respect to an invariant measure μ. The existence of μ is proven under suitable conditions on coefficients of the operator. We prove that the closure of N₀ is m-dissipative in      

A new perturbation to a critical elliptic problem with a variable exponent

In this paper, we study the following critical elliptic problem with a variable exponent:

$$\left\{ {\matrix{{ - \Delta u = {u^{p + \epsilon a\left( x \right)}}} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u > 0} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\partial \Omega ,} \hfill \cr } } \right.$$

where \(a\left( x \right) \in {C^2}\left( {\overline \Omega } \right),\,p = {{N + 2} \over {N - 2}},\,\,\epsilon > 0\), and Ω is a smooth bounded domain in ℝ^N (N ≽ 4). We show that for ∊ small enough, there exists a family of bubble solutions concentrating at the negative stable critical point of the function a(x). This is a new perturbation to the critical elliptic equation in contrast to the usual subcritical or supercritical perturbation, and gives the first existence result for the critical elliptic problem with a variable exponent.

     

On the Volterra integrodifferential equations and applications

For the generator A of a C ₀-semigroup on a Banach space (X, ∥·∥), we apply the perturbation of Desch-Schappacher type to solve the Volterra integordifferential equation VE $$\left\{ \begin{gathered} \frac{{du\left( t \right)}}{{dt}} = A\left( {u\left( t \right) + \int_0^t {a\left( {t - s} \right)B_1 u\left( s \right)ds + B_2 u\left( t \right) + B_3 f\left( t \right)} } \right) \hfill \\ + \int_0^t {b\left( {t - s} \right)B_4 u\left( s \right)ds + B_5 u\left( t \right) + g\left( t \right),t \geqslant 0,} \hfill \\ u\left( 0 \right) = u_0 , \hfill \\ \end{gathered} \right.$$ > which can be applied to treat boundary value problems and inhomogeneous retarded differential equations.     

Forward dynamical problem for the Timoshenko beam

The initial boundary value problem

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

Concentration phenomena in weakly coupled elliptic systems with critical growth*

In this paper we consider the weakly coupled elliptic system with critical growth

Smoothing properties of nonlinear transition semigroups: case of lipschitz nonlinearities

We consider a semilinear stochastic differential equation in a Hilbert space H with a Lipschitz continuous (possibly unbounded) nonlinearity F. We prove that the associated transition semigroup {P_t, t ≥ 0}, acting on the space of bounded measurable functions from H to , transforms bounded nondifferentiable functions into Fréchet differentiable ones. Moreover we consider the associated Kolmogorov equation and we prove that it possesses a unique “strong” solution (where “strong” means limit of classical solutions) given by the semigroup {P_t, t ≥ 0} applied to the initial condition. This result is a starting point to prove existence and uniqueness of strong solutions to Hamilton - Jacobi - Bellman equations arising in control theory. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

On a periodic prey-predator system with infinite delays

§1　IntroductionAnvarovandLarinov[1]introducedthefollowingprey-predatorsystem:x(t)=x(t)[α-γy(t)-γ∫∞0K1(s)y(t-s)ds-∫∞0∫∞0R1(s,θ)y(t-s)y(t-θ)dθds],y(t)=y(t)[-β μx(t) μ∫∞0K2(s)x(t-s)ds ∫∞0∫∞0R2(s,θ)x(t-θ)x(t-s)dθds],(1)whereα,γ,βandμarepositiveconstants,Ki∈C([0,∞),(0,∞))andRi∈C([0,∞)×[0,∞),(0,∞)),i=1,2.Fortheecologicalsenseofsystem(1),wereferto[1,2]andrefer-encescitedtherein.Sincerealisticmodelsrequiretheinclusionoftheeffectofchangingen-vironment,itmot…     

An end point estimate for maximal commutators

In this paper we prove that the maximal commutator of singular integral operator [b, T]^* satisfies the inequality:

Asymptotic behaviour of certain sets of associated prime ideals of Ext-modules

Let R be a commutative Noetherian ring, be an ideal of R and M be a finitely generated R-module. Melkersson and Schenzel asked whether the set becomes stable for a fixed integer i and sufficiently large j. This paper is concerned with this question. In fact, we prove that if s ≥ 0 and n ≥ 0 such that for all i with i < n, then is finite for all i with i < n, and is finite for all i with i ≤ n, where for a subset T of Spec(R), we set . Also, among other things, we show that if n ≥ 0, R is semi-local and is finite for all i with i < n, then is finite for all i with i ≤ n. K. Khashyarmanesh was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 86130027).