Two isoperimetric inequalities for the Sobolev constant

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.Comment: 11 page

Orbital stability of spherical galactic models

International audienceWe consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov in 1961. In a previous work, we derived the stability of anisotropic models under {\it spherically symmetric perturbations} using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics litterature. In this work, we show how this approach combined with a {\it new generalized} Antonov type coercivity property implies the orbital stability of spherical models under general perturbations

Nonlinear reinforcement problems with right-hand side in L1L^1

We study the asymptotic behaviour, of the entropy solution to a class of nonlinear ``reinforcement problems" and we find the "limit problem"

Isoperimetric inequalities in the parabolic obstacle problems

We are concerned with the parabolic obstacle problem ut+Au+cu≥f,u≥ψ, (ut+Au+cu−f)(u−ψ)=0inQ=(0,T)×Ω, u=ψ on Σ=(0,T)×∂Ω, u|t=0=u0 in Ω, A being a linear elliptic second-order operator in divergence form or a nonlinear `pseudo-Laplacian'. We give an isoperimetric inequality for the concentration of u−ψ around its maximum. Various consequences are given. In particular, it is proved that u−ψ vanishes after a finite time, under a suitable assumption on ψt+Aψ+cψ−f. Other applications are also given. "These results are deduced from the study of the particular case ψ=0. In this case, we prove that, among all linear second-order elliptic operators A having ellipticity constant 1, all equimeasurable domains Ω, all equimeasurable functions f and u0, the choice giving the `most concentrated' solution around its maximum is: A=−Δ, Ω is a ball Ω˜, f and u0 are radially symmetric and decreasing along the radii of Ω˜. "A crucial point in our proof is a pointwise comparison result for an auxiliary one-dimensional unilateral problem. This is carried out by showing that this new problem is well posed in L∞ in the sense of the theory of accretive operators