The Weighted Inertia-Energy-Dissipation Principle

TL;DR

The paper reviews the Weighted Inertia-Energy-Dissipation (WIDE) principle, a variational method for solving nonlinear evolution equations, highlighting its theoretical foundations, regularization properties, and applications.

Contribution

It provides a comprehensive survey of the WIDE variational approach, including its basic concepts, theoretical results, and practical applications across different types of differential equations.

Findings

WIDE functional offers elliptic-in-time regularization.

Limit process recovers solutions to nonlinear evolution equations.

Systematic review of theoretical and applied results.

Abstract

The Weighted Inertia-Energy-Dissipation (WIDE) principle is a global variational approach to nonlinear evolution equations of parabolic and hyperbolic type. The minimization of the parameter-dependent WIDE functional on trajectories delivers an elliptic-in-time regularization. By taking the limit in the parameter, one recovers a solution to the given differential problem. This survey is intended to provide a comprehensive account of the available results on the WIDE variational approach. The basic concepts are illustrated in the simplest finite-dimensional case, and the existing literature, both theoretical and applied, is systematically reviewed.