Minimizing non-convex multiple integrals: a density result

Abstract

We consider variational problems of the form where Ω is a bounded open set in R N , f : R N → R is a possibly non-convex lower semicontinuous function with p -growth at infinity for some 1 < p < ∞, and the boundary datum u 0 is any function in W 1, p (Ω). Assuming that the convex envelope f ** of f is affine on each connected component of the set { f ** < f }, we prove the existence of solutions to ( P ) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the set of continuous functions g that yield existence to ( P ) is dense in the space of continuous functions on R.