A note on approximation of optimal solutions of free problems of the calculus of variations

M. Lavrentiev, and later B. Manià, have given examples of free problems of the Calculus of Variations in which it is impossible to find a piecewiseC1 function, satisfying the same boundary conditions as the absolutely continuous optimal solution, and which simultaneously approximates both the optimal cost and, uniformly, the optimal trajectory. We investigate this question of approximation for free problems inEn.Dispensing with differentiability assumptions on the integrand of the cost functional, we present sufficient conditions under which this approximation problem may be solved thus extending earlier work of Manià, Tonelli, and others for free problems in the plane.