The Dirichlet integral as the model case. A general existence theorem: complete proof only in the case of differentiable integrands, with a reference to the general case of a Carathéodory function, convex in the gradient variable. The extension to the vectorial case: treatment of the special 2x2 case involving the determinant. An observation on non convex functionals using relaxation theory.

Reference: Chap. 3 of Dacorogna 2004 ICP book.