A multidimensional version of Liapunov-type theorems is proven. As an application, it is proven that, under proper hypothesis on the possibly nonconvex function f, the problem min integral(0)(T) f(u’(t)) dt on the subset of W-1,W-p([0,T],R(n)) Of those functions u satisfying the prescribed boundary conditions and whose trajectory lies out of a prescribed open subset of R(n) admits at least one solution.