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Nicolas Saintier

Publications and source records attributed to Nicolas Saintier.

27 records · Page 2Linked to original sources
arXiv2015

Metastability for small random perturbations of a PDE with blow-up

We study small random perturbations by additive space-time white noise of a reaction-diffusion equation with a unique stable equilibrium and solutions which blow up in finite time. We show that for initial data in the domain of attraction of the stable equilibrium the perturbed system exhibits metastable behavior: its time averages remain stable around this equilibrium until an abrupt and unpredictable transition occurs which leads to explosion in a finite (but exponentially large) time. On the other hand, for initial data in the domain of explosion we show that the explosion time of the perturbed system converges to the explosion time of the deterministic solution.

math.AP↗
arXiv2013

A Gamma convergence approach to the critical Sobolev embedding in variable exponent spaces

In this paper, we study the critical Sobolev embeddings $W^{1,p(x)}(Ω)\subset L^{p^*(x)}(Ω)$ for variable exponent Sobolev spaces from the point of view of the $Γ$-convergence. More precisely we determine the $Γ$-limit of subcritical approximation of the best constant associated with this embedding. As an application we provide a sufficient condition for the existence of extremals for the best constant.

math.AP↗
arXiv2013

On the Sobolev trace Theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev Trace Theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. Then we give local conditions on the exponents and on the domain (in the spirit of Adimurthy and Yadava) in order to satisfy such conditions, and therefore to ensure the existence of extremals.

math.AP↗
arXiv2013

Existence of solution to a critical trace equation with variable exponent

In this paper we study sufficient local conditions for the existence of non-trivial solution to a critical equation for the $p(x)-$Laplacian where the critical term is placed as a source through the boundary of the domain. The proof relies on a suitable generalization of the concentration--compactness principle for the trace embedding for variable exponent Sobolev spaces and the classical mountain pass theorem.

math.AP↗
arXiv2012

Existence of solution to a critical equation with variable exponent

In this paper we study the existence problem for the $p(x)-$Laplacian operator with a nonlinear critical source. We find a local condition on the exponents ensuring the existence of a nontrivial solution that shows that the Pohozaev obstruction does not holds in general in the variable exponent setting. The proof relies on the Concentration--Compactness Principle for variable exponents and the Mountain Pass Theorem.

math.AP↗
arXiv2011

On the Sobolev embedding theorem for variable exponent spaces in the critical range

In this paper we study the Sobolev embedding theorem for variable exponent spaces with critical exponents. We find conditions on the best constant in order to guaranty the existence of extremals. The proof is based on a suitable refinement of the estimates in the Concentration--Compactness Theorem for variable exponents and an adaptation of a convexity argument due to P.L. Lions, F. Pacella and M. Tricarico.

math.AP↗
arXiv2007

Estimates of the best Sobolev constant of the embedding of $BV(Ω)$ into $L^1(\partialΩ)$ and related shape optimization problems

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality $λ_1(Ω)\|u\|_{L^1(\partialΩ)} \le \|u\|_{W^{1,1}(Ω)}$ that are independent of $Ω$. This estimates generalize those of \cite{BS} concerning the $p$-Laplacian to the case $p=1$. We apply our results to prove existence of an extremal for this embedding. We then study an optimal design problem related to $λ_1$, and eventually compute the shape derivative of the functional $Ω\toλ_1(Ω)$. As a consequence, we obtain that a ball of $\R^n$ of radius $n$ is critical for volume-preserving deformations.

math.AP↗