Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and evolution equations

TL;DR

This paper extends variational principles based on anti-selfdual Lagrangians to unbounded, non self-adjoint operators, providing new formulations for complex evolution equations and boundary value problems.

Contribution

It introduces a variational framework for unbounded, non self-adjoint operators, expanding the applicability of ASD Lagrangians to more general evolution equations.

Findings

Developed variational formulations for unbounded operators

Applied methods to nonlinear boundary value problems

Addressed evolution equations with transport operators

Abstract

This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary equations of the form $ -Au\in \partial \phi (u)$ as well as i dissipative evolutions of the form $-\dot{u}(t)-A_t u(t)+\omega u(t) \in \partial \phi (t, u(t))$ were $\phi$ is a convex potential on an infinite dimensional space. In this paper, the emphasis is on the cases where the differential operators involved are not necessarily bounded, hence completing the results established in [13] for bounded linear operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by the transport operator with or without a diffusion term.