Hamiltonian systems of PDEs with selfdual boundary conditions

Abstract

Selfdual variational calculus is developed further and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, and Hamiltonian systems of PDEs. This allows for the resolution of such equations under general time boundary conditions which include the more traditional ones such as initial value problems, periodic and anti-periodic orbits, but also yield new ones such as "periodic orbits up to an isometry" for evolution equations that may not have periodic solutions. In the process, we introduce a method for perturbing selfdual functionals in order to induce coercivity and compactness, without destroying the selfdual character of the system.

A variational theory for monotone vector fields

“…• The class of anti-symmetric Hamiltonians that one can associate to selfdual Lagrangians ( [12], [19], [17], [18], [19]) goes beyond the theory of maximal monotone operators, and leads to a much wider array of applications. It shows among other things that they can be superposed with certain nonlinear operators that are far from being maximal monotone [12], [13] and [15].…”

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

“…As in Corollaries 4.8 and 4.10, by considering different combination of interior Nc-SD Lagrangians and boundary Nc-SD Lagrangians one can obtain different variational principles of Eq. (18). Here we state one more application of Theorem 4.7 and leave it to interested readers to generate more new principles by making use of Theorem 4.7.…”

“…Let ϕ : V → R and ψ : Y → R be convex,lower-semi continuous and also Gâteaux differentiable. If u is a critical point ofI (w) = 2ϕ * (Λw) − Λw, w + 2ψ * (β 2 w) − β 2 w, β 1 w then there exists v ∈ V such that v+u2 is a solution of(18).Proof. Define theNon-convex self-dual Lagrangians Φ : V × V * → R and : Y × Y * → R by Φ(u, p) = 2ϕ * (p) − u, p and (l, e) = 2ψ * (e) − e, l respectively.…”

Anti‐periodic solutions for a gradient system with resonance via a variational approach

“…To the best of our knowledge, few authors have studied the existence of anti‐periodic solutions for gradient systems by using variational approaches, especially for resonant anti‐periodic system. The readers refer to , , , , in which anti‐periodic solutions were studied by variational approaches. This paper is a generalization of applications of the dual least action principle.…”