Riemannian convexity of functionals

摘要

This paper extends the Riemannian convexity concept to action functionals defined by multiple integrals associated to Lagrangian differential forms on first order jet bundles. The main results of this paper are based on the geodesic deformations theory and their impact on functionals in Riemannian setting. They include the basic properties of Riemannian convex functionals, the Riemannian convexity of functionals associated to differential m-forms or to Lagrangians of class C~1 respectively C~2, the generalization to invexity and geometric meaningful convex functionals. The Riemannian convexity of functionals is the central ingredient for global optimization. We illustrate the novel features of this theory, as well as its versatility, by introducing new definitions, theorems and algorithms that bear upon the currently active subject of functionals in variational calculus and optimal control. In fact so deep rooted is the convexity notion that nonconvex problems are tackled by devising appropriate convex approximations.