Boundary value problems for degenerate elliptic equations and systems

Auscher, Pascal; Rosén, Andreas; Rule, David

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 951-1000.

On étudie les problèmes aux limites pour les équations et systèmes elliptiques dégénérés avec données au bord de carré intégrable. La dégénérescence est contrôlée par un poids $A_2$. On obtient représentation et trace pour les solutions dans des classes appropriées, des résultats de perturbation pour la résolubilité, et résolubilité pour certaines situations. La méthode de travaux antérieurs par les deux premiers auteurs est adaptée à ces dégénérescences une fois démontrée une estimation quadratique et nous améliorons même certains résultats dans le cas non-dégénéré. La preuve de cette estimation quadratique ne se déduit pas en revanche de résultats antérieurs et est la partie centrale de l'article.

We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an $A_2$ weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.

Classification : 35J56, 35J70, 35J25, 42B25, 42B37.
Keywords: Littlewood-Paley estimates, functional calculus, boundary value problems, second order elliptic equations and systems, weighted norm inequalities.
Mot clés : Estimées de Littlewood-Paley, calcul fonctionnel, problèmes aux limites, équations et systèmes elliptiques d'ordre deux, inégalités à poids.

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