Exponential stability of stochastic evolution equations driven by small fractional Brownian motion with Hurst parameter in (1/2,1)

This paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by Hölder continuous functions with Hölder index greater than 1/2. The results can be applied to the case of equations whose noisy inputs are given by a fractional Brownian motion $B^H$ with covariance operator Q, provided that $H\in (1/2,1)$ and $\mathrm{tr}(Q)$ is sufficiently small.     

Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d?2$, with boundary ?M, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T)\times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1)\cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ?M at fixed time. In the “flat” case where M is a compact subset of ${\mathbb{R}}^d$, two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation $\rho {?}_t^{\alpha }u?\mathrm{div}\left(a\mathrm{?}u\right)+qu=0$ on $(0,T)\times M$ are recovered simultaneously.     

Analysis of some injection bounds for Sobolev spaces by wavelet decomposition

We consider the Sobolev spaces $H^s(\Omega )$ and $H_0^s(\Omega )$ and the Besov spaces $B_{2,\infty }^{1/2}(\Omega )$, where Ω is a sufficiently regular (see Lemma 2) subdomain of ${\mathbb{R}}^2$. It is well known that for the values of $s\in [0,1/2)$ the two Sobolev spaces coincide, with equivalence of the norms, and that the inclusion $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ holds. The Note is concerned with the explicit analysis of the constants appearing in the continuity bounds for the injections $H^s(\Omega )?H_0^s(\Omega )$ and $B_{2,\infty }^{1/2}(\Omega )?H^s(\Omega )$ and of their dependence on the regularity s of the spaces. The analysis is carried out by using the wavelet characterization of the corresponding norms.     

Une propriété de composition dans l'espace Hs

Let us assume that $1<p?q?+\infty$, $1/p<s?[s]<1$, and $[s]?1$. If f and g are functions in the Besov space $B_p^{s\text{,}q}(\mathbb{R})$, such that g is real valued and such that $f(0)=0$, then the composed function $f○g$ belongs to $B_p^{s\text{,}q}(\mathbb{R})$. To cite this article: G. Bourdaud, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On the semiclassical approximation to the eigenvalue gap of Schrödinger operators

We consider two types of Schrödinger operators $H(t)=?d^2/d{x}^2+q(x)+t\cos x$ and $H(t)=?d^2/d{x}^2+q(x)+A\cos (tx)$ defined on $L^2(\mathbb{R})$, where q is an even potential that is bounded from below, A is a constant, and $t>0$ is a parameter. We assume that $H(t)$ has at least two eigenvalues below its essential spectrum; and we denote by ${\lambda }_1(t)$ and ${\lambda }_2(t)$ the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap $\Gamma (t)={\lambda }_2(t)?{\lambda }_1(t)$ in the limit as $t\to \infty$.     

Radial symmetry of positive solutions for semilinear elliptic equations in the unit ball via elliptic and hyperbolic geometry

Let $n\in \mathbb{N}$ with $n?2$, $a\in (?1,0)\cup (0,1]$ and $f:(0,1)\times (0,\infty )\to \mathbb{R}$ such that for each $u\in (0,\infty )$, $r?{(1+a{r}^2)}^{(n+2)/2}f(r,{(1+a{r}^2)}^{?(n?2)/2}u):(0,1)\to \mathbb{R}$ is nonincreasing. We show that each positive solution of$\mathrm{\Delta }u+f(|x|,u)=0\text{in}B,u=0\text{on}?B$ is radially symmetric, where B is the open unit ball in ${\mathbb{R}}^N$.     

Hölder estimates for fractional parabolic equations with critical divergence free drifts

This work focuses on drift-diffusion equations with fractional dissipation ${(?\mathrm{\Delta })}^{\alpha }$ in the regime $\alpha \in (1/2,1)$. Our main result is an a priori Hölder estimate on smooth solutions to the Cauchy problem, starting from initial data with finite energy. We prove that for some $\beta \in (0,1)$, the $C^{\beta }$ norm of the solution depends only on the size of the drift in critical spaces of the form $L_t^q({\mathrm{BMO}}_x^{?\gamma })$ with $q>2$ and $\gamma \in (0,2\alpha ?1]$, along with the $L_x^2$ norm of the initial datum. The proof uses the Caffarelli/Vasseur variant of De Giorgi's method for non-local equations.     

The coloring complex and cyclic coloring complex of a complete k-uniform hypergraph

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete k-uniform hypergraph. We show that the coloring complex of a complete k-uniform hypergraph is shellable, and we determine the rank of its unique nontrivial homology group in terms of its chromatic polynomial. We also show that the dimension of the $(n?k?1)\text{st}$ homology group of the cyclic coloring complex of a complete k-uniform hypergraph is given by a binomial coefficient. Further, we discuss a complex whose r-faces consist of all ordered set partitions $[B_1,\dots ,B_{r+2}]$ where none of the $B_i$ contain a hyperedge of the complete k-uniform hypergraph H and where $1\in B_1$. It is shown that the dimensions of the homology groups of this complex are given by binomial coefficients. As a consequence, this result gives the dimensions of the multilinear parts of the cyclic homology groups of $\mathbb{C}[x_1,\dots ,x_n]/\{x_{i_1}\dots x_{i_k}|i_1\dots i_k\text{is a hyperedge of}H\}$.     

Fractional parts of powers and Sturmian words

Let $b?2$ be an integer. In terms of combinatorics on words we describe all irrational numbers $\xi >0$ with the property that the fractional parts $\{\xi b^n\}$, $n?0$, all belong to a semi-open or an open interval of length $1/b$. The length of such an interval cannot be smaller, that is, for irrational ξ, the fractional parts $\{\xi b^n\}$, $n?0$, cannot all belong to an interval of length smaller than $1/b$. To cite this article: Y. Bugeaud, A. Dubickas, C. R. Acad. Sci. Paris, Ser. I 341 (2005).