EXISTENCE THEOREMS FOR A CLASS OF NONCONVEX PROBLEMS IN THE CALCULUS OF VARIATIONS

We give some existence results of minima for a class of nonconvex functionals depending on the Laplacian. We minimize these functionals on the set of functions u in W2,p(OMEGA) and W0(1,p)(OMEGA) such that partial derivative u/partial derivative n = 0 on partial derivative OMEGA, p > 1, with OMEGA either an annulus or the whole space R(n). Our approach allows us to deal with integrands without any regularity conditions. The results are obtained first by showing that the corresponding convexified problem has at least one radially symmetric solution via a rotation; then, by using a Liapunov's theorem on the range of a vector-valued measure, we construct a function that is a solution to our problem.