Existence of solutions to some classical variational problems

Some non-coercive variational integrals are considered, including the classical time-of-transit functional arising in the problem of the brachistochrone, and the area functional in the problem of the minimal surface of revolution. A minimizer is constructed by means of the direct method. More precisely, each admissible curve is replaced by its convex envelope, and the functional is shown to decrease. Hence, there exists a minimizing sequence made up of convex curves, which in turn possesses a locally uniformly converging subsequence. The limiting curve is a minimizer because the functionals under consideration are continuous under such a convergence.