Abstract:
A part of the calculus of variations deals with finding the minimum of integral functionals on the space of curves joining two points. The best known part of this theory is the necessary condition that any minimizer must satisfy the Euler-Lagrange equations for the problem. Lesser know, but possibly more important, is using the Hamilton-Jacobi equation to find extremals and using this equation to show extremals arising in this manner are in fact global minimizers. We show that this can proven easily using ideas from elementary convex analysis. In particular all that is needed is Young's inequality for conjugate convex function and the fundamental theorem of calculus. Hopefully this approach will allow for unified proofs of a variety of integral inequalities that at present have a rather diverse collection of proofs.