Analytical Results for Multifractal Properties of Spectra of Quasiperiodic Hamiltonians near the Periodic Chain

The multifractal properties of the electronic spectrum of a general quasiperiodic chain are studied in first order in the quasiperiodic potential strength. Analytical expressions for the generalized dimensions are found and are in good agreement with numerical simulations. These first order results do not depend on the irrational incommensurability.Comment: 10 Pages in RevTeX, 2 Postscript figure

Maximization of higher order eigenvalues and applications

The present paper is a follow up of our paper \cite{nS}. We investigate here the maximization of higher order eigenvalues in a conformal class on a smooth compact boundaryless Riemannian surface. Contrary to the case of the first nontrivial eigenvalue as shown in \cite{nS}, bubbling phenomena appear

Rigidity results for some boundary quasilinear phase transitions

We consider a quasilinear equation given in the half-space, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincar\'e inequality and, as a byproduct of this inequality, a result on the symmetry of low-dimensional bounded stable solutions, under some suitable assumptions on the nonlinearities. More precisely, we analyze the following boundary problem \left\{\begin{matrix} -{\rm div} (a(x,|\nabla u|)\nabla u)+g(x,u)=0 \qquad {on $\R^n\times(0,+\infty)$} -a(x,|\nabla u|)u_x = f(u) \qquad{\mbox{on $\R^n\times\{0\}$}}\end{matrix} \right. under some natural assumptions on the diffusion coefficient $a(x,|\nabla u|)$ and the nonlinearities $f$ and $g$. Here, $u=u(y,x)$, with $y\in\R^n$ and $x\in(0,+\infty)$. This type of PDE can be seen as a nonlocal problem on the boundary $\partial \R^{n+1}_+$. The assumptions on $a(x,|\nabla u|)$ allow to treat in a unified way the $p-$laplacian and the minimal surface operators

Besov algebras on Lie groups of polynomial growth

We prove an algebra property under pointwise multiplication for Besov spaces defined on Lie groups of polynomial growth. When the setting is restricted to the case of H-type groups, this algebra property is generalized to paraproduct estimates

Some elliptic PDEs on Riemannian manifolds with boundary

The goal of this paper is to investigate some rigidity properties of stable solutions of elliptic equations set on manifolds with boundary. We provide several types of results, according to the dimension of the manifold and the sign of its Ricci curvature

A note on precised Hardy inequalities on Carnot groups and Riemannian manifolds

We prove non local Hardy inequalities on Carnot groups and Riemannian manifolds, relying on integral representations of fractional Sobolev norms