An optimization problem with volume constraint for a degenerate quasilinear operator.

FERNÁNDEZ BONDER, JULIÁN; MARTÍNEZ, SANDRA RITA; WOLANSKI, NOEMI

Simposio; III International Symposium on Nonlinear PDEs and Free Boundary Problems; 2005

We consider the  optimization problem of minimizing$int_{Omega}| abla u|^p, dx$ with a constrain on the volume of${u>0}$. We consider a penalization problem, and we prove thatfor small values of the penalization parameter, the constrainedvolume is attained. In this way we prove that every solution $u$is locally Lipschitz continuous and that the free boundary,$partial{u>0}cap Omega$, is smooth.