Infima of universal energy functionals on homotopy classes

We consider the infima (f) on homotopy classes of energy functionals E defined on smooth maps f: Mⁿ → V^k between compact connected Riemannian manifolds. If M contains a sub‐manifold L of codimension greater than the degree of E then (f) is determined by the homotopy class of the restriction of f to M \ L. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semistability and finite maps

Let ${f : Y \longrightarrow M}Let f : Y ? M{f : Y \longrightarrow M} be a surjective holomorphic map between compact connected K?hler manifolds such that each fiber of f is a finite subset of Y. Let ω be a K?hler form on M. Using a criterion of Demailly and Paun (Ann. Math. 159 (2004), 1247–1274) it follows that the form f*ω represents a K?hler class. Using this we prove that for any semistable sheaf E ? M{E\, \longrightarrow\,M} , the pullback f*E is also semistable. Furthermore, f*E is shown to be polystable provided E is reflexive and polystable. These results remain valid for principal bundles on M and also for Higgs G-sheaves.     

P-convessità e convergenza di geodetiche

Summary We consider a sequence of energy functionals for regular paths with fixed extremes and whose range is contained in a corresponding sequence(M _h)_h∈Z+ of subsets of an Hilbert space. Assuming on eachM _h a condition similar top-convexity [C], we prove that if(M _h)_h∈Z+ is convergent in the sense of Kuratowsky toM the corresponding sequence(f _h)_h∈Z+of energy functionals is Γ-convergent to the functionalf relative toM and critical points off _h,i.e. the geodesics, are convergent to those off.      

Convergence of generalized gradients

For the graphs of Clarke's generalized gradients we prove that $$lim sup_{n \to + \infty } gph \partial f_n \subset gph \partial f in (E, strong) \times (E^* , weak).$$ provided that the sequencef _n of locally Lipschitz functions on a Banach spaceE with separable dual is strongly epi-convergent tof, equi-lower semidifferentiable and locally equibounded. This result extends [21] to the infinite-dimensional setting, and finds applications to the continuous behavior of the multiplier rule and of the generalized gradients of integral functionals under data perturbations.     

The Hahn-Banach Property and the Axiom of Choice

We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ? is a linear functional such that f ≤ p|F then there exists a linear functional g : E → ? such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

A Note on Sequential Properties of Noncompactness in Banach Spaces

We define two properties of sequences in Banach spaces that may be related to measures of noncompactness of subsets of these spaces. The first one concerns properties of sequences related to the strong topology, and the second one is related to the weak topology. Given a Banach space X, we introduce a new Banach space such that we can find a subset E in it that may be identified with the balls in the first one. We use compactness in this new space to characterize our sequential properties. In particular, we prove a general form of the Eberlein-Smulian theorem. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Gelfand-Phillips Property with Respect to the Weak Topology

We consider a Gelfand-Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E^?_? F inherits this property from E and F, and (2) the spaces L^p(μ, E) have this property when E does. A subset A of a Banach space E is a limited set if every (bounded linear) operator T:E → c₀ maps A onto a relatively compact subset of c₀. The Banach space E has the Gelfand-Phillips property if every limited set is relatively compact. In this note, we study the analogous notions set in the weak topology. Thus we say that A ? E is a Grothendieck set if every T: E → c₀ maps A onto a relatively weakly compact set; and E is said to have the weak type GP property if every Grothendieck set in E is relatively weakly compact. In the papers [3, 4 and 6], it is shown among other results that the ?-tensor product E and the spaces L^p(μ, E) inherit the Gelfand-Phillips property from E and F. In this paper, we study the same questions for the weak type GP property. It is easily verified that continuous linear images of Grothendieck sets are Grothendieck and that the weak type GP property is inherited by subspaces. Among the spaces with the weak type GP property one easily finds the separable spaces, and more generally, spaces with a weak* sequentially compact dual ball. Also, C(K) spaces where K is (DCSC) are weak type GP (see [3] and the discussion before Corollary 4 below). A Grothendieck space (a Banach space whose unit ball is a Grothendieck set) has the weak type GP if and only if it is reflexive.     

Smooth and strongly smooth points in symmetric spaces of measurable operators

We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? S_E^×{f\in S_{E^{\times}}} supporting μ(x), or s(x ^*) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.     

Some generic properties of α-nonexpansive mappings

Let X be a Banach space, C a bounded closed subset of X, A a convex closed subset of X, $\text{E}$ a complete metric space formed by all α-nonexpansive mappings fC → A and $\text{M}$ a complete metric space formed by α-nonexpansive differentiable mappings fC → X. The following assertions are proved in this paper: (1) Properness of I ? f is a generic property in $\text{E}$ (2)the subset of $\text{E}$ formed by all α-contractive mappings is of Baire first category in $\text{E}$; and (3) for every y?X, the functional equation x ? f(x) = y has generically a finite number of solutions for f in $\text{M}$. Some applications to the fixed point theory and calculation of the topological degree are given.     

On the Structure of the Solution Set of u″ = f(t,u), u(0) = u(1) = 0

The solution set of a Dirichlet problem x″ = f(t, x), x(0) = x(1) = 0, on a Banach space E and with f satisfying a Lipschitz condition, is homeomorphic to a closed subset of E. We prove that to an closed subset C of E there is a function f with Lipschitz constant arbitrarily close to π², such that the solution set of the corresponding Dirichlet problem is homeomorphic to C.     

Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces

As a consequence of the exposition of Dixmier type traces in the book of A. Connes (1994) [2], we were led to ask how general is this class of functionals within the space of all unitarily invariant functionals on the corresponding Marcinkiewicz ideal Mψ. In this paper we prove the surprising result that the set of all Dixmier traces on Mψ coincides with the set of all fully symmetric functionals on this space.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

Products and factors of Banach function spaces

Given two Banach function spaces we study the pointwise product space E · F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E · M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Hermitian Curvature and Plurisubharmonicity of Energy on Teichmüller Space

Let M be a closed Riemann surface, N a Riemannian manifold of Hermitian non-positive curvature, f : M → N a continuous map, and E the function on the Teichmüller space of M that assigns to a complex structure on M the energy of the harmonic map homotopic to f. We show that E is a plurisubharmonic function on the Teichmüller space of M. If N has strictly negative Hermitian curvature, we characterize the directions in which the complex Hessian of E vanishes.     

Problemi variazionali conformi

Letf: (M,g)→(N,g′) be a differentiable map between the riemannian manifoldsM andN, M being compact.K. Uhlenbeck pointed out a functionalE ^m(f), related to the energy density off, that depends only on the conformal structure ofM. In this paper we prove thatE ^m(f) is stationary with respect to deformations of the riemannian metric ofM if and only iff is weakly conformal; in this casef provides a local minimum ofE ^m.     

Maximization of Generalized Convex Functionals in Locally Convex Spaces

The major part of the investigation is related to the problem of maximizing an upper semicontinuous quasiconvex functional f over a compact (possibly nonconvex) subset K of a real Hausdorff locally convex space E. A theorem by Bereanu (Ref. 1) says that the condition f is quasiconvex (quasiconcave) on K is sufficient for the existence of maximum (minimum) point of f over K among the extreme points of K. But, as we prove by a counterexample, this is not true in general. On the further condition that the convex hull of the set of extreme points of K is closed, we show that it is sufficient to claim that f is induced-quasiconvex on K to achieve an equivalent conclusion. This new concept of quasiconvexity, which we define by requiring that each lower-level set of f can be represented as the intersection of K with some convex set, is suitable for functionals with a nonconvex domain. Under essentially the same conditions, we prove that an induced-quasiconvex functional f is directionally monotone in the sense that, for each y K, the functional f is increasing along a line segment starting at y and running to some extreme point of K. In order to guarantee the existence of maximum points on the relative boundary r K of K, it suffices to make weaker demands on the function f and the space E. By introducing a weaker kind of directional monotonicity, we are able to obtain the following result: If f is i.s.d.-increasing i.e., for each y y K, there is a half-line emanating from y such that f is increasing along this half-line, then f attains its maximum at rK , even if E is a topological linear Hausdorff space (infinite-dimensional and not necessarily locally convex). We state further a practical method of proving i.s.d.-monotonicity for functions in finite-dimensional spaces and we discuss also some aspects of classification.     

On the analyticity of CR mappings between nonminimal hypersurfaces

Let be a connected real-analytic hypersurface containing a connected complex hypersurface , and let be a smooth CR mapping sending M into another real-analytic hypersurface . In this paper, we prove that if f does not collapse E to a point and does not collapse M into the image of E, and if the Levi form of M vanishes to first order along E, then f is real-analytic in a neighborhood of E. In general, the corresponding statement is false if the Levi form of M vanishes to second order or higher, in view of an example due to the author. We also show analogous results in higher dimensions provided that the target M' satisfies a certain nondegeneracy condition. The main ingredient in the proof, which seems to be of independent interest, is the prolongation of the system defining a CR mapping sending M into M' to a Pfaffian system on M with singularities along E. The nature of the singularity is described by the order of vanishing of the Levi form along E. Received: 12 February 2001 / Published online: 18 January 2002     

A General Approximation Scheme for Attractors of Abstract Parabolic Problems

We consider semilinear problems of the form u′ = Au + f(u), where A generates an exponentially decaying compact analytic C ₀-semigroup in a Banach space E, f:E ^α → E is differentiable globally Lipschitz and bounded (E ^α = D((?A)^α) with the graph norm). Under a very general approximation scheme, we prove that attractors for such problems behave upper semicontinuously. If all equilibrium points are hyperbolic, then there is an odd number of them. If, in addition, all global solutions converge as t → ±∞, then the attractors behave lower semicontinuously. This general approximation scheme includes finite element method, projection and finite difference methods. The main assumption on the approximation is the compact convergence of resolvents, which may be applied to many other problems not related to discretization.