On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology _F on 2 ^X has as a subbase all sets of the form {A 2 ^X :A V 0}, whereV is an open subset ofX, plus all sets of the form {A 2 ^X :A W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for _F in terms of topological properties for . Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.     

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.     

Uniform Fat Segment and Cusp Properties for Compactness in Shape Optimization

A general compactness theorem for shape/geometric analysis and optimization is given for a family of subsets verifying the uniform fat segment property in a bounded open holdall with or without constraints on the De Giorgi [11] or the γ-density perimeter of Bucur and Zolesio [3]. The uniform fat segment property is shown to be equivalent to the uniform cusp property introduced in [9] with a continuous non-negative cusp function. This equivalence remains true for cusp functions that are only continuous at the origin. The equivalence of sets verifying a segment property with their C⁰-graph representation is further sharpened for sets with a compact boundary. Our C⁰-graph characterization is shown to be equivalent to both the uniform cusp property and the uniform segment property. It is used to formulate sufficient conditions on the local graphs of a family of subsets of a bounded open holdall to get compactness. A first condition assumes that the local graphs are bounded above by a cusp function; a second condition which requires that the local graphs be equicontinuous turns out to be equivalent to the first one. The respective solutions of the Laplacian with homogeneous Dirichlet or Neumann boundary condition are shown to be continuous with respect to domains in that family. In the Dirichlet case for 1 < p < ∞, we prove the (1,p)-stability of compact sets in the sense of Herdberg [14] under the weaker almost everywhere assumption rather than quasi everywhere. It is also shown that for the family of measurable crack free sets in a bounded open holdall implies .     

On closed convex sets without boundary rays and asymptotes

We study in finite-dimensional spaces the class of closed convex sets without boundary rays and asymptotes, denoted by and introduced by D. Gale and V. Klee. These sets, not necessarily bounded, enjoy many properties satisfied by compacts sets. New properties of this class are given and convergence analysis of this class is investigated. We also introduce the class of closed convex proper functions which have an epigraph in and we give some properties of these functions.     

Dominated Compactness Theorem in Banach Function Spaces and its Applications

A famous dominated compactness theorem due to Krasnosel’skiĭ states that compactness of a regular linear integral operator in L ^p follows from that of a majorant operator. This theorem is extended to the case of the spaces , with variable exponent p(·), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces is applied to fractional integral operators over bounded open sets. Submitted: June 6, 2007. Accepted: November 20, 2007.     

Epigraphical and Uniform Convergence of Convex Functions

We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of .

     

Convergence of difference schemes with high resolution for conservation laws

We are concerned with the convergence of Lax-Wendroff type schemes with high resolution to the entropy solutions for conservation laws. These schemes include the original Lax-Wendroff scheme proposed by Lax and Wendroff in 1960 and its two step versions-the Richtmyer scheme and the MacCormack scheme. For the convex scalar conservation laws with algebraic growth flux functions, we prove the convergence of these schemes to the weak solutions satisfying appropriate entropy inequalities. The proof is based on detailed estimates of the approximate solutions, compactness estimates of the corresponding entropy dissipation measures, and some compensated compactness frameworks. Then these techniques are generalized to study the convergence problem for the nonconvex scalar case and the hyperbolic systems of conservation laws.

     

Asymptotic behaviour of equicoercive diffusion energies in dimension two

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F _n , , defined in L ²(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F _n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F _n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.     

On algebras generated by convolutions and discontinuous functions

Banach algebras generated by Fourier and Mellin convolution operators with discontinuous presymbols and by discontinuous functions in L_p (IR⁺, x) spaces with weight are investigated. The Fredholm properties are characterized by a symbol calculus and an index formula for such operators is presented. These results were obtained by H. O. Cordes in [3] for the case p=2, =0 and presymbols, which are discontinuous only at infinity and generalized in [20] for 1相似文献   

On Two-Scale Convergence

In this paper we present the foundations of the two-scale convergence theory. We discuss the class of admissible functions in the mean-value formula and in the definition of two-scale convergence. We discuss the relation between strong and weak convergence and prove general theorems on the semicontinuity from below for convex functionals, and we also discuss the relation between two-scale convergence and monotonicity. The explanations are conducted on the level of L ^p -convergence; we do not deal with convergence in Sobolev spaces.     

Projections on Convex Sets in the Relaxed Limit

The paper establishes the continuity of the best approximation, or the projection, of a function in L _p for p[1,), on a closed convex set in the space, when the set varies and converges to a limit set in the Young-measure relaxation of the space. To this end a strong-type convergence and a convexity structure are identified on the space of Young measures. The appropriate convergence of sets with respect to which the continuity holds is the Mosco convergence of sets associated with the strong-type convergence of functions.     

Nonconforming finite element methods for eigenvalue problems in linear plate theory

The paper deals with nonconforming finite element methods for approximating fourth order eigenvalue problems of type ² w=w. The methods are handled within an abstract Hilbert space framework which is a special case of the discrete approximation schemes introduced by Stummel and Grigorieff. This leads to qualitative spectral convergence under rather weak conditions guaranteeing the basic properties of consistency and discrete compactness for the nonconforming methods. Further asymptotic error estimates for eigenvalues and eigenfunctions are derived in terms of the given orders of approximability and nonconformity. These results can be applied to various nonconforming finite elements used by Adini, Morley, Zienkiewicz, de Veubeke e.a. This is carried out for the simple elements of Adini and Morley and is illustrated by some numerical results at the end.     

On Positive Random Objects

The idea of defining the expectation of a random variable as its integral with respect to a probability measure is extended to certain lattice-valued random objects and basic results of integration theory are generalized. Conditional expectation is defined and its properties are developed. Lattice valued martingales are also studied and convergence of sub- and supermartingales and the Optional Sampling Theorem are proved. A martingale proof of the Strong Law of Large Numbers is given. An extension of the lattice is also studied. Studies of some applications, such as on random compact convex sets in R ⁿ and on random positive upper semicontinuous functions, are carried out, where the generalized integral is compared with the classical definition. The results are also extended to the case where the probability measure is replaced by a -finite measure.     

Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension.” Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces W _p ¹ on the “zero” cusp with a Hölder singularity at the vertex.     

Über den Auswahlsatz von Blaschke und ein Problem von Valentine

Following an idea of Marjanovi [9] we prove an extensive generalization of the Blaschke convergence theorem. Using similar methods we are able to give the solution of a problem concerning the compactness of families of starshaped sets in a topological linear space, which was posed by Valentine [11].     

Measures of weak noncompactness in Banach spaces

Measures of weak noncompactness are formulae that quantify different characterizations of weak compactness in Banach spaces: we deal here with De Blasi's measure ω and the measure of double limits γ inspired by Grothendieck's characterization of weak compactness. Moreover for bounded sets H of a Banach space E we consider the worst distance k(H) of the weak^∗-closure in the bidual of H to E and the worst distance ck(H) of the sets of weak^∗-cluster points in the bidual of sequences in H to E. We prove the inequalities

     

Right Markov Processes and Systems of Semilinear Equations with Measure Data

In the paper we prove the existence of probabilistic solutions to systems of the form ?Au = F(x, u) + μ, where F satisfies a generalized sign condition and μ is a smooth measure. As for A we assume that it is a generator of a Markov semigroup determined by a right Markov process whose resolvent is order compact on L¹. This class includes local and nonlocal operators corresponding to Dirichlet forms as well as some operators which are not in the variational form. To study the problem we introduce new concept of compactness property relating the underlying Markov process to almost everywhere convergence. We prove some useful properties of the compactness property and provide its characterization in terms of Meyer’s property (L) of Markov processes and in terms of order compactness of the associated resolvent.     

Toeplitz and Hankel type operators on the upper half-plane

An orthogonal decomposition of admissible wavelets is constructed via the Laguerre polynomials, it turns to give a complete decomposition of the space of square integrable functions on the upper half-plane with the measurey dxdy. The first subspace is just the weighted Bergman (or Dzhrbashyan) space. Three types of Ha-plitz operators are defined, they are the generalization of classical Toeplitz, small and big Hankel operators respectively. Their boundedness, compactness and Schatten-von Neumann properties are studied.Research was supported by the National Natural Science Foundation of China.     

On two-scale convergence and related sequential compactness topics

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L ²(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.     

Supremum metric and relatively compact sets of fuzzy sets

The supremum metric D between fuzzy subsets of a metric space is the supremum of the Hausdorff distances of the corresponding level sets. In this paper some new criteria of compactness with respect to the distance D are given; they concern arbitrary fuzzy sets (see Theorem 7), fuzzy sets having no proper local maximum points (see Theorem 12) and, finally, fuzzy sets with convex sendograph (see Theorem 13). In order to compare results with a previous characterization of compactness of Diamond–Kloeden, the criteria will be expressed by equi-(left/right)-continuity. In the proofs a first author's purely topological criterion of D  -compactness and a variational convergence (called Γ$\Gamma$-convergence) which was introduced by De Giorgi and Franzoni, are fundamental.