A VARIATIONAL PROBLEM FOR HARMONIC-FUNCTIONS IN RING-SHAPED DOMAINS WITH PARTIALLY FREE-BOUNDARY

This paper considers two subsets OMEGA0 and OMEGA of R(n), n = 2 or n = 3, and two continuous real-valued functions go and g defined on partial derivative OMEGA and partial derivative OMEGA0, respectively. The position of OMEGA is allowed to vary inside OMEGA0 and the author looks for the minimum of the Dirichlet intergral of the function u, which is harmonic in (OMEGA0\OMEGA) and verifies the following boundary conditions: u = g0 on partial derivative OMEGA0, u = g on partial derivative OMEGA. Under certain hypotheses on the regularity of partial derivative OMEGA0 and partial derivative OMEGA, and on g0 and g, an existence theorem is proved for the minimizing position of OMEGA; it is shown through an example that the solution of the considered problem is not unique in general.