Global strong solution to the Thirring model in critical space

We obtain a strong solution in charge critical space L²(R) of the Thirring system and Federbusch equations in one space dimension by using solution representation of the models. The uniqueness is obtained for the solution Ψ∈L^∞([0,T];L²(R)∩L⁴(R)). A decay of local charge and asymptotic behavior of the field can be shown directly.     

Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems

By computing the E-critical groups at θ and infinity of the corresponding functional of Hamiltonian systems, we proved the existence of nontrivial periodic solutions for the systems which may be resonant at θ and infinity under some new conditions. Some results in the literature are extended and some new type of theorems are proved. The main tool is the E-Morse theory developed by Kryszewski and Szulkin.     

Multiple solutions for some nonlinear Schrödinger equations with indefinite linear part

We consider a class of nonlinear Schrödinger equation with indefinite linear part in RN. We prove that the problem has at least three nontrivial solutions by means of Linking Theorem and (∇)-Theorem.     

A note on ball-covering property of Banach spaces

By a ball-covering B of a Banach space X, we mean that B is a collection of open (or closed) balls off the origin whose union contains the unit sphere SX of X; and X is said to have the ball-covering property (BCP) provided it admits a ball-covering by countably many balls. In this note we give a natural example showing that the ball-covering property of a Banach space is not inherited by its subspaces; and we present a sharp quantitative version of the recent Fonf and Zanco renorming result saying that if the dual X^∗ of X is w^∗ separable, then for every ε>0 there exist a (1+ε)-equivalent norm on X, and an R>0 such that in this new norm SX admits a ball-covering by countably many balls of radius R. Namely, we show that R=R(ε) can be taken arbitrarily close to (1+ε)/ε, and that for X=?₁[0,1] the corresponding R cannot be equal to 1/ε. This gives the sharp order of magnitude for R(ε) as ε→0.     

Variations on Weyl's theorem

In this note we study the property (w), a variant of Weyl's theorem introduced by Rako?evi?, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T^*) coincide whenever T^* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).     

The length of a shortest geodesic net on a closed Riemannian manifold

In this paper we will estimate the smallest length of a minimal geodesic net on an arbitrary closed Riemannian manifold Mⁿ in terms of the diameter of this manifold and its dimension. Minimal geodesic nets are critical points of the length functional on the space of immersed graphs into a Riemannian manifold. We prove that there exists a minimal geodesic net that consists of m geodesics connecting two points p,q∈Mⁿ of total length ≤md, where m∈{2,…,(n+1)} and d is the diameter of Mⁿ. We also show that there exists a minimal geodesic net with at most n+1 vertices and geodesic segments of total length .These results significantly improve one of the results of [A. Nabutovsky, R. Rotman, The minimal length of a closed geodesic net on a Riemannian manifold with a nontrivial second homology group, Geom. Dedicata 113 (2005) 234-254] as well as most of the results of [A. Nabutovsky, R. Rotman, Volume, diameter and the minimal mass of a stationary 1-cycle, Geom. Funct. Anal. 14 (4) (2004) 748-790].     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

Hardy spaces for the strip

In this paper we shall study Hardy spaces of analytic functions in a strip S. Our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense. We also present an orthogonal (in H²(S)) basis of polynomials.     

Dirichlet and VMOA domains via Schwarzian derivative

Short proofs of the following results concerning a bounded conformal map g of the unit disc D are presented: (1) logg^′ belongs to the Dirichlet space if and only if the Schwarzian derivative Sg of g satisfies Sg(z)(1−²|z|)∈L²(D); (2) logg^′∈VMOA if and only if ²|Sg(z)|³(1−²|z|) is a vanishing Carleson measure on D. Analogous results for Besov and Qp,0 spaces are also given.     

On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L¹-spaces and prove that every surjective isometry from the unit sphere of L¹(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L¹(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L^∞(Γ), C₀(Ω) and L^∞(μ)), then every bijective ?-isometry between the unit spheres of E and F can be extended to a bijective 5?-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.     

Local regularity theorems for the stationary thermistor problem with oscillating degeneracy

In this paper we present some regularity results for solutions to the system −Δu=σ(u)²|∇φ|, div(σ(u)∇φ)=0 in the case where σ(u) is allowed to oscillate between 0 and a positive number as u→∞. In particular, we show that u is locally bounded if σ(u) is bounded below by a suitable exponential function.     

Functional calculus and ∗-regularity of a class of Banach algebras II

In this article, we define a natural Banach ∗-algebra for a C^∗-dynamical system (A,G,α) which is slightly bigger than L¹(G;A) (they are the same if A is finite-dimensional). We will show that this algebra is ∗-regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and ∗-regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press].     

On discretizations of invariant foliations over inertial manifolds

Invariant foliations over inertial manifolds of partial differential equations under numerical discretizations are studied. It is proved that the numerical method considered as a discrete dynamical system has C¹-close invariant foliations. The rate of the C¹-convergence is estimated as well.     

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u^′(t)+B(t,u(t))∋f(t) is studied. In this problem, the operators are of type (M) or type (S₊), which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.     

Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory

Let X be a real reflexive Banach space with dual X^∗. Let L:X⊃D(L)→X^∗ be densely defined, linear and maximal monotone. Let T:X⊃D(T)→X^∗2, with 0∈D(T) and 0∈T(0), be strongly quasibounded and maximal monotone, and C:X⊃D(C)→X^∗ bounded, demicontinuous and of type (S₊) w.r.t. D(L). A new topological degree theory has been developed for the sum L+T+C. This degree theory is an extension of the Berkovits-Mustonen theory (for T=0) and an improvement of the work of Addou and Mermri (for T:X→X^∗2 bounded). Unbounded maximal monotone operators with are strongly quasibounded and may be used with the new degree theory.     

Characterization of equality in a generalized Dunkl-Williams inequality

We establish a generalization of the Dunkl-Williams inequality and its inverse in the framework of Hilbert C^?-modules and characterize the equality case. As applications, we get some new results and some known results due to J.E. Pe?ari? and R. Raji? [The Dunkl-Williams equality in pre-Hilbert C^?-modules, Linear Algebra Appl. 425 (1) (2007) 16-25].     

Banach space characterizations of unitaries: A survey

We survey Banach space characterizations of unitary elements of C^∗-algebras, JB^∗-triples, and JB-algebras. In the case of the existence of a pre-dual, appropriate specializations of these characterizations are also reviewed.     

Multilinear commutators of Calderón-Zygmund operators on Hardy-type spaces with non-doubling measures

Under the assumption that μ is a non-negative Radon measure on Rd which only satisfies some growth condition, the authors obtain the boundedness in some Hardy-type spaces of multilinear commutators generated by Calderón-Zygmund operators or fractional integrals with RBMO(μ) functions, where the Hardy-type spaces are some appropriate subspaces, associated to the considered RBMO(μ) functions, of the Hardy space H¹(μ) of Tolsa.     

Remarks on invariant metrics on Teichmüller space

We make some comparisons concerning the induced infinitesimal Kobayashi metric, the induced Siegel metric, the L² Bergman metric, the Teichmüller metric and the Weil-Petersson metric on the Teichmüller space of a compact Riemann surface of genus g?2. As a consequence, among others, we show that the moduli space has finite volume with respect to the L² Bergman metric. This answers a question raised by Nag in 1989.     

Exact boundary observability for nonautonomous quasilinear wave equations

By means of a direct and constructive method based on the theory of semiglobal C² solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous wave equations and autonomous ones is also revealed.