On the necessary condition of Weierstrass in the multiple integral problem of the calculus of variations III

Abstract

curves minimizing an integral ff(x, y, y') dx has been established by Tonelli under hypotheses which could hardly be weakened. The extension to curves in space has not been carried out, to the best of my knowledge; but it seems that it should be possible without great difficulty. On the other hand, the analogous condition for the multiple integral problem has by no means been established with the same generality. Levi' considered