p(x)-harmonic functions with unbounded exponent in a subdomain
José Miguel Urbano
Departamento de Matemática
Universidade de Coimbra

Abstract

We study the Dirichlet problem for the p(x)-Laplacian, in the case of a variable exponent p(x) that is infinite in a subdomain D. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit of the solutions u_n to the corresponding problem when p_n(x) = min (p(x),n), in particular, with p_n=n in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem, which is, in addition, infinity-harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it solves.

(Joint work with Juan J. Manfredi and Julio D. Rossi.)

Jornada SPM/CIM/CMAT - "Dia das Equações"