Nonconvex Minimization Problems for Functionals Defined on Vector Valued Functions

We consider the minimization problem min H fŽ Žx.. hŽ Žx.. dx, W01, 1ŽBRn, m.BRn where BRn is the ball of n centered at the origin and with radius R 0, f is a lower semicontinuous function, and h is a convex function. We give sufficient conditions for the existence and uniqueness of minimizers. Our technique relies on a detailed knowledge of the properties of the solutions to the convexified problem, obtained using the corresponding Euler Lagrange inclusions.