Well-posed elliptic Neumann problems involving irregular data and domains

Alvino, Angelo; Cianchi, Andrea; Maz'ya, Vladimir G.; Mercaldo, Anna

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 4, pp. 1017-1054.

Nous considérons des problèmes de Neumann pour des équations elliptiques non linéaires dans des domaines éventuellement non réguliers et avec des données peu régulières. Un équilibre entre l'intégrabilité de la donnée et l'(ir)régularité du domaine nous permet d'obtenir l'existence, l'unicité et la dépendance continue de solutions généralisées. L'irrégularité du domaine est décrite par des inégalités « isocapacitaires ». Nous donnons aussi des applications à certaines classes de domaines.

Non-linear elliptic Neumann problems, possibly in irregular domains and with data affected by low integrability properties, are taken into account. Existence, uniqueness and continuous dependence on the data of generalized solutions are established under a suitable balance between the integrability of the datum and the (ir)regularity of the domain. The latter is described in terms of isocapacitary inequalities. Applications to various classes of domains are also presented.

Classification : 35J25, 35B45
Mots clés : Non-linear elliptic equations, Neumann problems, Generalized solutions, A priori estimates, Stability estimates, Capacity, Perimeter, Rearrangements

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