Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.     

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u   is a solution of (−Δ)^su=g${(-\mathrm{\Delta })}^{s}u=g$ in Ω  , u≡0$u\equiv 0$ in Rⁿ\Ω${\mathbb{R}}^n\backslash \Omega$, for some s∈(0,1)$s\in (0,1)$ and g∈L^∞(Ω)$g\in L^{\infty }(\Omega )$, then u   is C^s(Rⁿ)$C^s({\mathbb{R}}^n)$ and u/δ^s_|Ω$u/{\delta }^s{|}_{\Omega }$ is C^α$C^{\alpha }$ up to the boundary ∂Ω   for some α∈(0,1)$\alpha \in (0,1)$, where δ(x)=dist(x,∂Ω)$\delta (x)=\mathrm{dist}(x,\partial \Omega )$. For this, we develop a fractional analog of the Krylov boundary Harnack method.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

A semilinear elliptic problem on unbounded domains with reverse penalty

In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)u^p-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?_b∞?_lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<_b∞$b(x)<b_{\infty }$. For any periodic lattice L   in R^N${\mathbb{R}}^N$ and for any b∈C(R^N)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<_b∞$b(x)<b_{\infty }$, _b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination b^Y$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=b^Y(x)u^p-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$.     

Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Δu+K(|x|)u^p=0$\mathrm{\Delta }u+K(|x|)u^p=0$ in R^N${\mathbf{R}}^N$ for N>2$N>2$ and p>1$p>1$, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r^−?K(r)$r^{-?}K(r)$ with ?>−2$?>-2$ around a positive constant is small near r=∞$r=\infty$ and p   is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent _pc$p_c$ which is determined by N   and the order of the behavior of K(r)$K(r)$ as r=|x|→0$r=|x|\to 0$ and ∞. In order to understand how subtle the structure is on K   at p=_pc$p=p_c$, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2)$p=(N+2)/(N-2)$.     

p-Laplacian boundary value problems on exterior regions

This paper treats some variational principles for solutions of inhomogeneous p  -Laplacian boundary value problems on exterior regions U?R^N$U?{\mathbb{R}}^N$ with dimension N?3$N?3$. Existence-uniqueness results when p∈(1,N)$p\in (1,N)$ are provided in a space E^1,p(U)$E^{1,p}(U)$ of functions that contains W^1,p(U)$W^{1,p}(U)$. Functions in E^1,p(U)$E^{1,p}(U)$ are required to decay at infinity in a measure theoretic sense. Various properties of this space are derived, including results about equivalent norms, traces and an L^p$L^p$-imbedding theorem. Also an existence result for a general variational problem of this type is obtained.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

Characterizing graphic matroids by a system of linear equations

Given a rank-r   binary matroid we construct a system of O(r³)$O(r^3)$ linear equations in O(r²)$O(r^2)$ variables that has a solution over GF(2)$\mathrm{GF}(2)$ if and only if the matroid is graphic.     

Derivations and local derivations on Freudenthal almost f-algebras

Let A be an Archimedean f  -algebra and let N(A)$N(A)$ be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A$d:A\to A$ is a derivation if and only if d(A)⊂N(A)$d(A)\subset N(A)$ and d(A²)={0}$d(A^2)=\{0\}$, where A²$A^2$ is the set of all products ab in A.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

Approximation in weighted p-mean by RBF networks of Delsarte translates

We give conditions on the kernel function (or activation function) for the family of radial basis function (RBF) neural networks obtained upon replacing the usual translation by the Delsarte one, with not necessarily the same smoothing factor in all kernel nodes, to have the universal approximation property in suitable weighted L^p$L^p$-spaces (1?p<∞)$(1?p<\infty )$. A complete characterization of such kernels for p=1$p=1$ and p=2$p=2$ is provided.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.     

From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A  , in a two-dimensional domain with L²$L^2$ cells. For fixed A  , and L→∞$L\to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when L   is fixed, and A→∞$A\to \infty$. In this case the solution equilibrates along stream lines.     

Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α=1/2$\alpha =1/2$) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L²$L^2$ to L^∞$L^{\infty }$, from L^∞$L^{\infty }$ to Hölder (Cδ$C^{\delta }$, δ>0$\delta >0$), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L^∞$L^{\infty }$, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L^∞$L^{\infty }$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α$C^{1-2\alpha }$. Higher regularity starting from Cδ$C^{\delta }$ with δ>1−2α$\delta >1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].     

Hyperspaces of Keller compacta and their orbit spaces

A compact convex subset K   of a topological linear space is called a Keller compactum if it is affinely homeomorphic to an infinite-dimensional compact convex subset of the Hilbert space _?2${?}_2$. Let G be a compact topological group acting affinely on a Keller compactum K   and let 2^K$2^K$ denote the hyperspace of all non-empty compact subsets of K endowed with the Hausdorff metric topology and the induced action of G  . Further, let cc(K)$cc(K)$ denote the subspace of 2^K$2^K$ consisting of all compact convex subsets of K. In a particular case, the main result of the paper asserts that if K   is centrally symmetric, then the orbit spaces 2^K/G$2^K/G$ and cc(K)/G$cc(K)/G$ are homeomorphic to the Hilbert cube.