A variational approach to symmetry, monotonicity, and comparison for doubly-nonlinear equations

TL;DR

This paper introduces a variational method based on the Weighted-Energy-Dissipation approach to establish qualitative properties like symmetry and comparison principles for a broad class of doubly-nonlinear evolutionary PDEs, including nonlocal and fractional problems.

Contribution

It develops a novel variational framework that simplifies proving qualitative properties for complex nonlinear PDEs by analyzing minimizers of a global functional.

Findings

Proves symmetry and monotonicity for solutions of doubly-nonlinear equations.

Establishes comparison principles via the WED variational approach.

Extends the method to rate-independent and hyperbolic systems.

Abstract

We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter goes to zero. Qualitative properties and comparison principles can be easily proved for minimizers of the WED functional and, by passing to the limit, for the limiting problem. Several applications of the abstract results to systems of nonlinear PDEs and to fractional/nonlocal problems are presented. Eventually, we present some extensions of this approach in order to deal with rate-independent systems and hyperbolic problems.