Proceedings of the Royal Society of Edinburgh: Section A Mathematics



On some Bernoulli free boundary type problems for general elliptic operators


J. I. Díaz a1, J. F. Padial a2 and J. M. Rakotoson a3
a1 Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Plaza de Ciencias 3, Ciudad Universitaria, 28040 Madrid, Spain (ildefonso.diaz@mat.ucm.es)
a2 Departamento de Matemática Aplicada, ETSA, Universidad Politécnica de Madrid, Avenida Juan de Herrera 4, 28040 Madrid, Spain
a3 Laboratoire de Mathématiques, SP2MI, Université de Poitiers, Boulevard Marie et Pierre Curie, Téléport 2, BP30179, 86962 Futuroscope Chasseneuil Cedex, France

Article author query
diaz ji   [Google Scholar] 
padial jf   [Google Scholar] 
rakotoson jm   [Google Scholar] 
 

Abstract

We consider some Bernoulli free boundary type problems for a general class of quasilinear elliptic (pseudomonotone) operators involving measures depending on unknown solutions. We treat those problems by applying the Ambrosetti–Rabinowitz minimax theorem to a sequence of approximate nonsingular problems and passing to the limit by some a priori estimates. We show, by means of some capacity results, that sometimes the measures are regular. Finally, we give some qualitative properties of the solutions and, for a special case, we construct a continuum of solutions.

(Received March 27 2006)
(Accepted July 12 2006)