Non-convex self-dual Lagrangians: New variational principles of symmetric boundary value problems

Abstract

We study the concept and the calculus of Non-convex self-dual (Nc-SD) Lagrangians and their derived vector fields which are associated to many partial differential equations and evolution systems. They indeed provide new representations and formulations for the superposition of convex functions and symmetric operators. They yield new variational resolutions for large class of Hamiltonian partial differential equations with variety of linear and nonlinear boundary conditions including many of the standard ones. This approach seems to offer several useful advantages: It associates to a boundary value problem several potential functions which can often be used with relative ease compared to other methods such as the use of EulerLagrange functions. These potential functions are quite flexible, and can be adapted to easily deal with both nonlinear and homogeneous boundary value problems. Additionally, in most cases the solutions generated using this new method have greater regularity than the solutions obtained using the standard Euler-Lagrange function. Perhaps most remarkable, however, are the permanence properties of Nc-SD Lagrangians; their calculus is relatively manageable, and their applications are quite broad.

A variational principle for problems with a hint of convexity

“…We shall briefly recall some of them and refer the interested reader to [12] where some more general versions of Theorem 1.2 are established and several applications in the fixed point theory and PDEs are provided. It is also worth noting that Theorem 1.2 extends some of variational principles established by the author in [10,11]. We shall now proceed with some applications.…”

Non-convex self-dual Lagrangians and variational principles for certain PDEʼs

“…It was then understood that all variational principles of this type fall under a unified principle as discussed in a series of papers [3][4][5]. One can indeed use this corollary (see [4]) to provide an existence result for system of super-linear transport equations with a small parameter ,…”

Multiplicity results for elliptic problems with super-critical concave and convex nonlinearties

“…We shall be proving Theorems 1.1 and 1.2 by making use of a new abstract variational principle established recently in [13,14] (see also [11,12] for some new variational principles and [5] for an application in supercritical Neumann problems). To be more specific, let V be a reflexive Banach space, V * its topological dual and let K be a convex and weakly closed subset of V .…”