TITLE

Noncoercive convectionâ€“diffusion elliptic problems with Neumann boundary conditions

AUTHOR(S)
Droniou, Jérôme; Vázquez, Juan-Luis
PUB. DATE
April 2009
SOURCE
Calculus of Variations & Partial Differential Equations;Apr2009, Vol. 34 Issue 4, p413
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
We study the existence and uniqueness of solutions of the convectiveâ€“diffusive elliptic equation posed in a bounded domain $$\Omega\subset {\mathbb{R}}^N$$ , with pure Neumann boundary conditions Under the assumption that $$V\in L^p(\Omega)^N$$ with p = N if N â‰¥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution $$u\in H^1(\Omega)$$ if âˆ«Î© f dx = 0, and also that the kernel is generated by a function $$\widehat{u} \in H^1(\Omega)$$ , unique up to a multiplicative constant, which satisfies $$\widehat{u} > 0$$ a.e. on Î©. We also prove that the equationhas a unique solution for all Î½ > 0 and the map $$f \mapsto u$$ is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure data and to parabolic problems.
ACCESSION #
36336454

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