The existence of minimizers of the action functional without convexity assumption

Abstract

We shall prove the existence of minimizers of the following functional $f(u)=\int_{0}^{T}L(x,u(x),u'(x))dx$ without convexity assumption. As a consequence of this result and the duality described in [A. Nowakowski, Metody wariacyjne dla nieliniowych problem\'ow Dirichleta (Chapter 6), Wydawnictwo Naukowo Techniczne, Warszawa, 1994] we derive the existence of solutions for the Dirichlet problem for a certain differential inclusion being a generalization of the Euler-Lagrange equation of the functional $f$.