A direct method for doubly nonlinear equations via convexification in spaces of measures and duality

TL;DR

This paper introduces a direct variational method for doubly nonlinear equations in measure spaces, avoiding time discretization and leveraging convexification and duality principles to establish solution existence.

Contribution

It develops a novel, direct approach based on convexification and duality in measure spaces, bypassing traditional iterative schemes for doubly nonlinear equations.

Findings

Proves existence of solutions via a measure-based variational approach.

Identifies the dual problem as a Hamilton-Jacobi inequality.

Extends the method to non-autonomous equations with time-dependent potentials.

Abstract

Existence of solutions to doubly nonlinear equations in reflexive Banach spaces is established by resorting to a global-in-time variational approach inspired by De Giorgi's principle, which characterizes the associated flows as null-minimizers of a suitable energy-dissipation functional defined on trajectories. In contrast to the celebrated minimizing movements scheme, the proposed strategy does not rely on any time-discretization or iterative constructions. Instead, it provides a direct method based on the relaxation of the problem in spaces of measures, constrained by the continuity equation: in this procedure, no gap is introduced due to the Ambrosio's superposition principle. Within this weak convex framework, the validity of the null-minimization property is recovered through two further steps. First, a careful application of the Von Neumann minimax theorem yields an identification of the dual problem as a supremum over the set of smooth and bounded cylinder functions, solving an Hamilton-Jacobi-type inequality. Secondly, a suitable "backward boundedness" property of solutions to such Hamilton-Jacobi system gives a proper bound of the dual problem, ensuring that the minimum value of the original functional is actually zero. The proposed strategy naturally extends to non-autonomous equations, encompassing time- and space-dependent dissipation potentials and time-dependent potential energies.