Schrodinger equations and Hamiltonian systems of PDEs with selfdual boundary conditions

TL;DR

This paper refines selfdual variational calculus to establish existence results for solutions of PDEs and Hamiltonian systems under various boundary conditions, including novel types like 'periodic orbits up to an isometry.'

Contribution

It introduces a method for perturbing selfdual functionals to ensure coercivity and compactness while maintaining selfduality, enabling broader solution existence results.

Findings

Established existence of solutions under general boundary conditions.

Introduced a perturbation method for selfdual functionals.

Extended solution concepts to 'periodic orbits up to an isometry'.

Abstract

Selfdual variational calculus is further refined and used to address questions of existence of local and global solutions for various parabolic semi-linear equations, Hamiltonian systems of PDEs, as well as certain nonlinear Schrodinger evolutions. 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, while keeping the system selfdual.