Non-convex self-dual Lagrangians and new variational principles of symmetric boundary value problems: Evolution case

Abstract

In this work, we shall present some new variational principles for evolutionary equations by the virtue of the Non-convex self-dual (Nc-SD) Lagrangians. It is established that how lifting Nc-SD Lagrangians to path spaces allows one to associate to an evolution boundary value problem several potential functions, which can often be used with relative ease compared to other methods such as the use of Euler-Lagrange functionals. These Lagrangians, indeed provide new representations and formulations for the superposition of semi-convex functions and symmetric operators. They yield new variational resolutions for large class of hamiltonian partial differential equations with a variety of linear and nonlinear boundary conditions including many of the standard ones. They can be adapted to easily deal with both nonlinear and homogeneous boundary value problems and, in most cases, solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function.