Positive quasi-minima
We study the existence and nonexistence of positive solutions of nonlinear elliptic systems in an annulus with Dirichlet boundary conditions. In particular, a priori bounds are obtained. We also study a general multiple linear eigenvalue problem on a bounded domain.
In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere . We derive from this analysis some a priori estimates in dimension and . On these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On we prove the existence of at least one...
We consider sequences of solutions of the Navier-Stokes equations in , associated with sequences of initial data bounded in . We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in , up to a remainder term small in ; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If is an “admissible” space (in particular ...
In this article, we consider the set F of functions annihilated by a uniformly elliptic system S in an open set Ω of Rn.We show that, as in the case of harmonic functions, F satisfies a submean-property, first for p=2 by elliptic estimates, then for all p > 0:|∇k u(x)|p ≤ C / (rn+kp) ∫B(x,r) |u(y)|p dyfor each u in F, each k > 0 and every ball B(x,r) included in Ω.As a consequence, we can compare ||u||Lp(Ω) and ||∇ku||Lp(Ω,δkp) where δ is the distance to the boundary of Ω, under the...
We study in this paper a notion of pseudo-spectrum in the semi-classical setting called injectivity pseudo-spectrum. The injectivity pseudo-spectrum is a subset of points in the complex plane where there exist some quasi-modes with a precise rate of decay. For that reason, these values can be considered as some ‘almost eigenvalues’ in the semi-classical limit. We are interested here in studying the absence of injectivity pseudo-spectrum, which is characterized by a global a priori estimate. We prove...