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This paper is concerned with the regularity, forward-compactness and measurability of pullback random attractors of non-autonomous stochastic lattice systems with multiplicative white noise as well as random coefficients in weighted spaces. The existence, uniqueness and forward-compactness of pullback random attractors in weighted space l σ 2 are proved for the systems when the random coefficient in the front of the discrete Laplacian and the time-dependent functions satisfy certain conditions. In addition, this forward-compact ( l σ 2 , l σ 2 ) -attractor is proved to be a forward-compact bi-spatial ( l σ 2 , l σ 2 ∩ l σ p ) -attractor that is forward-compact and measurable in l σ 2 ∩ l p σ , and attracts all random subset of l σ 2 under the topology of l σ 2 ∩ l σ p . The difficulty of establishing the measurability of the attractors in l σ 2 ∩ l σ p is overcome by proving the identity of the attractors on two different universes.

In this paper, we construct compact Toeplitz operators and trace class Toeplitz operators with unbounded symbols on Hardy-Sobolev spaces. We also characterize the compactness of finite sums of products of two Toeplitz operators on these spaces. Then, we study the Fredholmness of the Toeplitz operators and calculate the essential norm of them. In addition, we consider the C ⁎ -algebra generated by Toeplitz operators.

In this work we provide compactness results which are generalizations of Baras' Theorem for the case when the main operator is time-dependent and we give applications for nonautonomous coupled inclusions, proving that these multivalued problems define exact generalized processes.

Let ( K , d ) be a compact metric space, 0 α ≤ 1 and Lip α K the space of the Lipschitz functions on K. It is known that the Banach algebra Lip α K is a BSE-algebra. In this paper, for a commutative unital semisimple Banach algebra A , we prove that the Banach algebra Lip α ( K , A ) of the A -valued Lipschitz functions is a BSE-algebra if and only if A is so.

Suppose that 〈 f n 〉 is a sequence of polynomials, 〈 f n ( k ) ( 0 ) 〉 converges for every non-negative integer k, and that the limit is not 0 for some k. It is shown that if all the zeros of f 1 , f 2 , … lie in the closed upper half plane Im z ≥ 0 , or if f 1 , f 2 , … are real polynomials and the numbers of their non-real zeros are uniformly bounded, then the sequence converges uniformly on compact sets in the complex plane. The results imply a theorem of Benz and a conjecture of Polya.

This paper considers the initial-boundary value problem for the nonlocal Cahn–Hilliard equation ∂ t φ + ( − Δ + 1 ) ( a ( ⋅ ) φ − J ⁎ φ + G ′ ( φ ) ) = 0 in ( 0 , T ) × Ω in an unbounded domain Ω ⊂ R N with smooth bounded boundary, where N ∈ N , T > 0 , and a ( ⋅ ) , J , G are given functions. In the case that Ω is a bounded domain and − Δ + 1 is replaced with −Δ, this problem has been studied by using a Faedo–Galerkin approximation scheme considering the compactness of the Neumann operator − Δ + 1 (cf. [8] , [16] ). However, the compactness of the Neumann operator − Δ + 1 breaks down when Ω is an unbounded domain. The present work establishes existence and energy estimates of weak solutions for the above problem on an unbounded domain.

Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu asymmetric excess between compact sets from the target metric space. For this, we study subtle properties of the directional variations. We show by examples that all assumptions in the main existence result are essential. As an application, we establish the existence of set-valued solutions $X(t)$ of bounded variation to the functional inclusion of the form $X(t)\subset F(t,X(t))$ satisfying the initial condition $X(t_0)=X_0$., 32 pages, LaTeX, uses elsarticle

In the paper we formulate a criterion for relative compactness in the space R ( J , E ) consisting of all regulated functions, defined on the compact interval J and with values in the Banach space E. On the basis of the criterion we construct two arithmetically convenient and regular measures of noncompactness and investigate the connections of these measures with the Hausdorff measure. We show that obtained estimates are optimal. Moreover, we formulate necessary and sufficient conditions for the Nemytskii operator to be acting from the space R ( J , E ) into itself. Finally, we show the applicability of the mentioned measures in proving the existence of solutions of a nonlinear functional-integral equation.

We study the topological entropy h ( f ) of continuous endomorphisms f of compact-like groups. More specifically, we consider the e-spectrum E t o p ( K ) for a compact-like group K (namely, the set of all values h ( f ) , when f runs over the set E n d ( K ) of all continuous endomorphisms of K). We pay particular attention to the class E ∞ of topological groups without continuous endomorphisms of infinite entropy (i.e., ∞ ∉ E t o p ( K ) ) as well as the subclass E 0 of E ∞ consisting of those groups K with E t o p ( K ) = { 0 } . It turns out that the properties of the e-spectrum and these two classes are very closely related to the topological dimension. We show, among others, that a compact connected group K with finite-dimensional commutator subgroup belongs to E ∞ if and only if dim ⁡ K ∞ and we obtain a simple formula (involving the entropy function) for the dimension of an abelian topological group which is either locally compact or ω-bounded (in particular, compact). Examples are provided to show the necessity of the compactness or commutativity conditions imposed for the validity of these results (e.g., compact connected semi-simple groups K with dim ⁡ K = ∞ and K ∈ E 0 , or countably compact connected abelian groups with the same property). Since the class E ∞ is not stable under taking closed subgroups or quotients, we study also the largest subclasses S ( E ∞ ) and Q ( E ∞ ) , respectively, of E ∞ , having these stability properties. We provide a complete description of these two classes in the case of compact groups, that are either abelian or connected. The counterpart for S ( E 0 ) and Q ( E 0 ) is done as well.

Charges are linear functionals on normal chains in a compact metric space that are continuous with respect to a modified flat norm topology. They define a cohomology which reflects metric, rather than topological, properties of the underlying space. We show that charges can be represented by metric differentiable forms build from tuples of Lipschitz functions, and that the representation is unique on the cohomology level. For specific forms, the wedge products and the products with normal chains define cup and cap products, respectively.

We consider the Banach function space consisting of real functions defined on the half-axis and having increments tempered by a given modulus of continuity. Next, we formulate and prove a sufficient condition for relative compactness of a bounded subset of the mentioned Banach space. On the basis of that condition we construct a measure of noncompactness in the function space in question. Using that measure we prove a theorem on the existence of solutions of a quadratic integral equation in the space of functions satisfying the Holder condition on the real half-axis.

Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space H 1 over finite-dimensional real space are generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases and examples of compact Lie groups, homogeneous groups (in particular the Heisenberg group) and finite-dimensional spaces over division rings are considered. In conclusion, we solve for the space L 2 an open question on compactness of a Hausdorff operator posed by Liflyand.

Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M . Let t τ be the measure topology on the ⁎-algebra S ( M , τ ) of all τ-measurable operators. We prove that for B ∈ S ( M , τ ) + the sets I B = { A ∈ S ( M , τ ) h : − B ≤ A ≤ B } and K B = { A ∈ S ( M , τ ) : A ⁎ A ≤ B } are convex and t τ -closed in S ( M , τ ) . In this case, we have I B = { B T B : T ∈ M h and ‖ T ‖ ≤ 1 } and, for invertible B, we describe the set of extreme points of the set I B . Let M be an atomic von Neumann algebra. We prove that an operator B ∈ S ( M , τ ) + is τ-compact if and only if the set I B is t τ -compact. The t τ -compactness of I B for all τ-compact operators B characterizes these algebras.

The paper is devoted to the existence of weak solutions of set-valued stochastic differential equations. Such equations were considered in the authors paper [22] , where existence and uniqueness of strong solutions have been investigated. Now, apart from the existence of weak solutions, the weak compactness in distribution of the set of all weak solutions of set-valued stochastic differential equations, is considered.

A bivariate extreme-value copula is characterized by a function of one variable, called a Pickands dependence function, which is convex and comprised between two bounds. The authors identify the smallest possible compact set containing the graph of all Pickands dependence functions whose corresponding bivariate extreme-value copula has a fixed value of Spearman's rho or Kendall's tau. The consequences of this result for statistical modeling are outlined.

International audience; Druet [6] proved that if $(f_\gamma)_\gamma$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_\gamma)_\gamma$ solves $$ \begin{cases}&\Delta u_\gamma =f_\gamma(x,u_\gamma)\,,~~ u_\gamma>0\text{ in }\Omega\,,\\&u_\gamma =0\text{ on }\partial\Omega\,,\end{cases}$$ and converges weakly in $H^1_0$ to some $u_\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\ge 0$ such that the energy of $u_\gamma$ converges to $4\pi N$ plus the Dirichlet energy of $u_\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in [8] that, for a specific class of nonlinearities, the loss of compactness (i.e. $N>0$) implies that $u_\infty\equiv 0$. In contrast, we prove here that there exist sequences $(f_\gamma)_\gamma$ of Moser-Trudinger type nonlinearities which admit a noncompact sequence $(u_\gamma)_\gamma$ of solutions having a nontrivial weak limit.

In this paper, we are interested in weak solutions of the following Schrodinger–Kirchhoff equation involving double critical nonlinearities: { − ( a + b ∫ R 3 | ∇ u | 2 ) Δ u = u 2 ⁎ ( s 1 ) − 1 | x | s 1 + u 2 ⁎ ( s 2 ) − 1 | x | s 2 , in R 3 ∖ { 0 } , u ∈ D 1 , 2 ( R 3 ) , u ≥ 0 in R 3 , where a , b are two positive constants, 2 ⁎ ( s ) = 2 ( 3 − s ) and 0 ≤ s 1 s 2 2 . A natural strategy is to search the weak solutions of the equation as critical points of a suitable functional via variational methods. But by dilation invariance we see that the functional doesn't satisfy ( P S ) condition. We will make use of several techniques to recover compactness. Our results show that the higher order term dominates lower order term and there exist multi-scale characteristics in this nonlinear problem.

For a vector measure m, with values in a Banach space, we analyze the localization of uniformly integrable subsets of scalar integrable functions with respect to m in a suitable Orlicz space associated to m. As tools we consider the properties of Orlicz spaces associated to intermediate functions of a given pair of Young functions. These properties allow us to obtain compactness properties of the inclusion operators.

On a Boolean algebra we consider the topology $u$ induced by a finitely additive measure $\mu$ with values in a locally convex space and formulate a condition on $u$ that is sufficient to guarantee the convexity and weak compactness of the range of $\mu$. This result a la Lyapunov extends those obtained in (Khan, Sagara 2013) to the finitely additive setting through a more direct and less involved proof. We will then give an economical interpretation of the topology $u$ in the framework of coalitional large economies to tackle the problem of measuring the bargaining power of coalitions when the commodity space is infinite dimensional and locally convex. We will show that our condition on $u$ plays the role of the "many more agents than commodities" condition introduced by Rustichini and Yannelis in (1991). As a consequence of the convexity theorem, we will obtain two straight generalizations of Schmeidler's and Vind's Theorems on the veto power of coalitions of arbitrary economic weight.

We introduce the notion of topological r-entropy for free semigroup actions on a compact metric space and provide some properties of it. By using the skew-product transformation as bridge, we get the following two main results. 1. We extend the result that the topological entropy is the limit of topological r-entropy in [15] to free semigroup actions ( r → 0 ). 2. Let f i , i = 0 , 1 , ⋯ , m − 1 , be homeomorphisms on a compact metric space. We verify that the topological entropy of f 0 , ⋯ , f m − 1 equals the topological entropy of f 0 − 1 , ⋯ , f m − 1 − 1 .

We establish that C p ( X ) has a dense subspace of countable i-weight if and only if d ( C p ( X ) ) ⩽ c . It is also proved that the space C p ( K ) has a dense F σ -discrete subspace whenever K is Corson compact. If both spaces X and C p ( X ) are Lindelof Σ, then for any natural n ⩾ 2 , the iterated function space C p , n ( X ) has a uniformly dense subspace of countable pseudocharacter. In the case of a Gul'ko compact space K, there is a uniformly dense subspace of countable pseudocharacter in C p , n ( K ) for any n ∈ N . This is a new result even for C p ( K ) given that K is Eberlein compact.

The famous Weierstrass theorem asserts that every continuous function on a compact set in R d can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varju which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.

The main goal of this work is to prove the existence of three different solutions (one positive, one negative and one with nonconstant sign) for the equation $(-\Delta_p)^s u= |u|^{p^{*}_s -2} u +\lambda f(x,u)$ in a bounded domain with Dirichlet condition, where $(-\Delta_p)^s$ is the well known $p$-fractional Laplacian and $p^*_s=\frac{np}{n-sp}$ is the critical Sobolev exponent for the non local case. The proof is based in the extension of the Concentration Compactness Principle for the $p$-fractional Laplacian and Ekeland's variational Principle., Comment: 10 pages. arXiv admin note: text overlap with arXiv:0808.3143, arXiv:0912.3465

We first prove that the set of shadowable measures of a homeomorphism of a compact metric space X is an F σ δ set of the space M ( X ) of Borel probability measures of X equipped with the weak*-topology. Next that the set of shadowable measures is dense in M ( X ) if and only if the set of shadowable points is dense in X. Therefore, if X has no isolated points and every non-atomic Borel probability measure has the shadowing property, then the shadowable points are dense in X (this is false when the space has isolated points). Afterwards, we consider the almost shadowable measures (measures for which the shadowable point set has full measure) and prove that all of them are weak* approximated by shadowable ones. In addition the set of almost shadowable measures is a G δ set of M ( X ) . Furthermore, the closure of the shadowable points is the union of the supports of the almost shadowable measures. Finally, we prove that every almost shadowable measure can be weak* approximated by ones with support equals to the closure of the shadowable points.

The current paper contemplates the blowup dynamics in trapped dipolar quantum gases. More precisely, employing the profile decomposition of bounded sequences in H ˙ 1 ∩ H ˙ 1 2 , we firstly construct related variational problems and derive two refined Gagliardo–Nirenberg inequalities. Secondly, a compactness lemma is utilized to prove that the blowup solutions with bounded H ˙ 1 2 norm and bounded L 3 norm absolutely concentrate at least a fixed amount.

Dugundji spaces were introduced by Pelczynski as compact Hausdorff spaces X such that every embedding of X into a Tychonoff cube [ 0 , 1 ] A admits a linear extension operator u : C ( X ) → C ( [ 0 , 1 ] A ) such that ‖ u ‖ = 1 and u ( 1 X ) = 1 [ 0 , 1 ] A , where 1 X is the constant function on X taking value 1. In this paper we show that a compact space X is Dugundji provided that there exists a linear extension operator u : C ( X ) → C ( [ 0 , 1 ] A ) satisfying one of the following conditions: (a) ‖ u ‖ 2 and | u ( f ⋅ g ) | ≤ ‖ g ‖ ⋅ | u ( | f | ) | for all f , g ∈ C ( X ) ; (b) ‖ u ‖ = 1 .

In the paper [10] the L p -realization L p of the matrix Schrodinger operator L u = d i v ( Q ∇ u ) + V u was studied. The generation of a semigroup in L p ( R d , C m ) and characterization of the domain D ( L p ) has been established. In this paper we perturb the operator L p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on L p ( R d , C m ) with domain embedded in W 2 , p ( R d , C m ) . We also study the analyticity, compactness, positivity and ultracontractivity of the semigroup and prove Gaussian kernel estimates. Further kernel estimates and asymptotic behaviour of eigenvalues of the matrix Schrodinger operator are investigated.

We consider a mathematical model which describes the motion of a 3D unsteady fluid flow governed by the Navier–Stokes system, and subjected to mixed boundary conditions with a given velocity on one part of the boundary and nonlinear slip conditions with a memory term reminiscent of Coulomb's friction law on the other part. We establish first some regularity properties and estimates for a simplified model. Then we prove the existence of a solution to our problem by using a successive approximation technique and compactness arguments based on Helly's theorem for the velocity field.

We consider a hemivariational inequality of elliptic type (HVI, for short) in a reflexive Banach space, prove its solvability and the compactness of its set of solutions. To this end we employ a surjectivity theorem for multivalued mappings that we use for the sum of a maximal monotone operator and a bounded pseudomonotone operator. Next, we introduce the concepts of strongly and weakly well-posedness in the generalized sense for the HVI and provide two characterizations for the strongly well-posedness, under different assumptions on the data. These characterizations are formulated in terms of the metric properties of the approximating sets. We also provide sufficient conditions which guarantee the weakly and strongly well-posedness in the generalized sense of the HVI. Finally, we consider two perturbations of the HVI for which we obtain convergence results in the sense of Kuratowski.

The aim of this paper is to prove Kolmogorov-Riesz type theorems via Bessel and Laguerre translations, and Pego-type theorems by the corresponding transformations.

We supply two optimal lower bounds for the solution of the analytical traveling salesman problem in terms of Jones' β-numbers. We prove a linear lower bound for the length of every connecting curve in the general case. The four corners Cantor set shows that the linear bound is attained. For connected compact sets, we prove an exponential lower bound for their length. The estimate is optimal by the work of Bishop and Jones [2] . Finally, we bridge connected and disconnected cases by taking into account multiplicity of orthogonal projections. Applications to other areas of mathematics are briefly discussed.

The compactness and the stability of the weak solution of nonautonomous maximal monotone flows are proved with respect to nonparametric perturbations of the operator. To this purpose the flow is formulated in weak form as a null-minimization problem, and De Giorgi's notion of Γ-convergence is used. It is proved that all sequences of those flows have a Γ-convergent subsequence, and that the Γ-limit is also a maximal monotone flow and exhibits no long memory. These results can be applied to several quasilinear equations.

We completely characterize the boundedness and compactness of Volterra type integration operators J b between Hardy-Carleson type tent spaces CT p , α . Furthermore, we show that the compactness and strict singularity of J b : CT p , α → CT p , α are equivalent for all 1 ≤ p ∞ and α > − n − 1 . Our tools mainly involve the atomic decomposition for the space CT p , α , Khinchine's inequality and multipliers of some sequence tent spaces. Corresponding results are also obtained for the little spaces CT p , α 0 .

In this note, we give some applications of projective logarithmic potentials. First we introduce the notions of projective logarithmic energy and capacity associated to projective kernel. We compare quantitatively the projective logarithmic capacity with the complex Monge-Ampere capacity on complex projective space and we deduce that the set of zero logarithmic capacity is of Monge-Ampere capacity zero. Further, we define transfinite diameter of a compact set and we show that it coincides with logarithmic capacity. Finally we deduce that there is an analogous of classical Evans's theorem.

Let X be a compact metric space and let C ( Y ) be the space of all complex-valued continuous functions on a Hausdorff compact space Y. We prove that the isometry group of the algebra Lip ( X , C ( Y ) ) of all C ( Y ) -valued Lipschitz maps on X, equipped with the sum norm, is topologically reflexive and 2-topologically reflexive whenever the isometry group of C ( Y ) is topologically reflexive. The same results are established for the sets of isometric reflections and generalized bi-circular projections of Lip ( X ) .

We consider a Markov decision process with countable state space and Borel action space. We are interested in maximizing the probability that the controlled Markov chain visits some subset of the state space infinitely often. We provide sufficient conditions, based on continuity and compactness requirements, together with a stability condition on a parametrized family of auxiliary control models, which imply the existence of an optimal policy that is deterministic and stationary. We compare our hypotheses with those existing in the literature.

The existence of solutions to a fourth-order p-Laplacian equation with boundary degeneracy is studied. For the purpose of solving the corresponding non-degenerate (with respect to the coefficient of fourth-order term) regularized problem, a fourth-order semi-discrete elliptic problem with homogeneous boundary conditions is established and its existence and uniqueness are obtained by the functional minimization method. It follows that the approximate solutions of the non-degenerate parabolic problem are constructed and the corresponding existence and uniqueness are discovered by a limit procedure from the energy estimation method and a compactness argument. Finally, the existence and regularity of solutions for the problem with boundary degeneracy is obtained by using a regularization parameter vanishing limit.

We consider a family of $*$-commuting local homeomorphisms on a compact space, and build a compactly aligned product system of Hilbert bimodules (in the sense of Fowler). This product system has a Nica-Toeplitz algebra and a Cuntz-Pimsner algebra. Both algebras carry a gauge action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in this torus. We study the KMS states of these dynamics. For large inverse temperatures, we describe the simplex of KMS states on the Nica-Toeplitz algebra. To study KMS states for smaller inverse temperature, we consider a preferred dynamics for which there is a single critical inverse temperature. We find a KMS state on the Nica-Toeplitz algebra at this critical inverse temperature which factors through the Cuntz-Pimsner algebra. We then illustrate our results by considering backward shifts on the infinite-path spaces of a class of $k$-graphs., Comment: To appear in JMAA

In this paper we prove the existence of a nontrivial τ-antisymmetric solution for the following system { − Δ u + u = | u | 2 p − 2 u + β ( x ) | v | p | u | p − 2 u , in R N − Δ v + ω 2 v = | v | 2 p − 2 v + β ( x ) | u | p | v | p − 2 v , in R N where N ≥ 3 , 2 2 p 2 ⁎ , ω > 0 is a positive constant and τ : R N → R N is a nontrivial orthogonal involution. Here β is a continuous function, invariant under τ, satisfying additional conditions, some of them related to ω. The basic tool employed here is the Concentration Compactness Principle.

In this paper, we propose and study a multi-type SIS nonlocal epidemic model with seasonality. We first establish the existence of the spreading speed c ⁎ and the non-existence of periodic traveling waves with wave speed c c ⁎ . Then, we prove the existence and asymptotic behavior of the periodic traveling fronts with speed c > c ⁎ . Finally, the existence of the critical periodic traveling wave fronts is established by using a limiting argument. To overcome the difficulty of lack of compactness of solution maps of the nonlocal system with respect to compact open topology, we show that the solution sequence is pre-compact in L loc ( R 2 , R m ) .

The aim of this note is to discuss the relation between the assumptions of the Poincare–Miranda theorem and the viability condition, first used by Nagumo to prove existence of a solution to ODEs under state constraints (viable solutions). An interesting consequence of this observation is an extension of the Poincare–Miranda theorem to arbitrary convex compact sets in locally convex Hausdorff vector spaces (instead of a parallelotope in an Euclidean space). We also recall a very short proof of this extension based on a Ky Fan inequality. This proof is not new, but seems to have passed unnoticed in the literature devoted to the Poincare–Miranda theorem. In fact, recent variations of this theorem in l 2 follow then in a simple straightforward way. The above extension also implies a generalization of the Lax intermediate value theorem to infinite dimensional spaces.

Let H = { z ∈ C : Im z > 0 } be the upper half plane, and denote by L p ( R ) , 1 ≤ p ∞ , the usual Lebesgue space of functions on the real line R. We define two “composition operators” acting on L p ( R ) induced by a Borel function φ : R → H ‾ , by first taking either the Poisson or Borel extension of f ∈ L p ( R ) to a function on H ‾ , then composing with φ and taking vertical limits. Classical composition operators, induced by holomorphic functions and acting on the Hardy spaces H p ( H ) of holomorphic functions, correspond to a special case. Our main results provide characterizations of when the operators we introduce are bounded or compact on L p ( R ) , 1 ≤ p ∞ . The characterization for the case 1 p ∞ is independent of p and the same for the Poisson and the Borel extensions. The case p = 1 is quite different.

We study the existence of solutions for the nonlinear Schrodinger equation − Δ u + V ( x ) u = f ( x , u ) in R 2 , where the potential V is 1-periodic, 0 lies in a spectral gap from the spectrum of the Schrodinger operator S = − Δ + V and the nonlinearity f ( x , t ) has exponential growth in the sense of Trudinger–Moser. The main feature here is that f ( x , t ) is allowed to be both periodic and nonperiodic in the x variable. Our proofs rely on a linking theorem and the Lions concentration compactness principle.

As it is well-known, f ∈ R R is continuous if and only if f maps continua (compact, connected sets) to continua. The same holds for mappings between any two (real or complex) normed spaces. However, when we restrict ourselves to polynomials P : E → K , where E is a K -normed space, then it was proved in 2012 that P is continuous if and only if it transforms compact sets into compact sets. Here we show that (if K = C ) P is continuous if and only if it transforms connected sets into connected sets. Although we provide some partial results for K = R , the general case in the real setting remains still open.

We consider the L ∞ weak solutions to a type of compressible Euler equation with dissipation effects. Several studies [6] , [14] , [30] have obtained the L ∞ weak solutions to this type of system by using numerical schemes and the compensated compactness method. Therefore, the uniform boundedness of approximate solutions and the H l o c − 1 compactness of the corresponding entropy dissipation measures must be considered. It should be noted that the obtained L ∞ bounds typically increase over time. However, getting a time-independent uniform bound is important to consider the large time behavior of weak solutions. In this paper, by using invariant region theory, we prove that the L ∞ weak solutions derived by the Lax–Friedrichs scheme are uniformly bounded in time.

For $0, Comment: 23 Pages

Kato [5] and Artigue [3] merged the theory of expansive systems [10] and foliations with the continuum theory [14] . Here we merge the expansive systems but with the descriptive set theory [6] instead. More precisely, we define meagre-expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively. We first prove that every cw-expansive homeomorphism of a locally connected metric space without isolated points is meagre-expansive (but not conversely). Second that a homeomorphism of a metric space is meagre-expansive if and only if every Borel probability measure is meagre-expansive. Next that the space of meagre-expansive measures of a homeomorphism of a compact metric space X is an F σ subset of the space of Borel probability measures equipped with the weak* topology. In the sequel we prove that every homeomorphism with a meagre-expansive measure of a compact metric space has an invariant meagre-expansive measure. Also that the set of periodic points of every meagre-expansive homeomorphism of a compact metric space has empty interior. In the circle or the interval we prove that there are no meagre-expansive homeomorphisms of the circle or the interval. Moreover, the meagre-expansive measures of an interval homeomorphism or a circle homeomorphism with rational rotation number are precisely the finite convex combinations of Dirac measures supported on isolated periodic points. A circle homeomorphism with irrational rotation number has a meagre-expansive measure if and only if it is a Denjoy map. In such a case the meagre-expansive measures are precisely those measures supported on the unique minimal set of the map. To obtain some of our results we will consider a measurable version of the classical Baire Category.

Under the right conditions on a compact metric space X and on a Banach space E, we give a description of the 2-local (standard) isometries on the Banach space Lip ( X , E ) of vector-valued Lipschitz functions from X to E in terms of a generalized composition operator, and we study when every 2-local (standard) isometry on Lip ( X , E ) is both linear and surjective.

An inverse problem for a system of semilinear elliptic equations modeling simultaneously conductive and radiative heat transfer is under consideration. The problem consists in finding the right-hand side of the heat transfer equation, in the form of linear combination of given functionals, on the base of prescribed values of these functionals on the solution. The solvability of the problem is proven without any smallness assumptions. It is shown that the set of solutions is homeomorphic to a finite-dimensional compact set, and conditions of uniqueness of solutions are found.