The optimization of a functional within the class of solutions of a partial differential equation

Summary This paper deals with a class of variational problems involving multiple integrals with one unknown function. Else than in the classical calculus of variations where the unknown function e.g. must be continuous and must take fixed values on the boundary, the unknown function must be a solution of a partial differential equation.Physically one could imagine a process, described by a partial differential equation and controlled by the boundary conditions, while this process must be optimized in some sense by choosing the best boundary values.