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

Alessandro Pinzi, Filippo Riva, Giuseppe Savaré

TL;DR

The paper develops a direct, global-in-time method to establish existence of solutions for doubly nonlinear evolution equations $\partial\psi(\dot x) + \partial\varphi(x) \ni 0$ in reflexive Banach spaces by a De Giorgi–type energy-dissipation principle. It relaxes the problem to spaces of Banach-valued measures under a continuity equation, and employs a Von Neumann minimax approach to obtain a dual Hamilton–Jacobi formulation, together with a backward-boundedness argument to bound the dual and yield a zero minimum of the primal functional. The framework avoids time discretization and is robust to nonautonomous dissipation and time-dependent energies, with a structured three-step proof: convexification in measure spaces, duality reduction to a Hamilton–Jacobi constraint, and verification of a comparison-type principle ensuring zero dual-minimum. This provides a versatile existence theory for a broad class of rate-dependent dissipative systems and offers a flexible variational lens for analyzing doubly nonlinear flows under nonconvex energies and nonautonomous settings.

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.

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

TL;DR

The paper develops a direct, global-in-time method to establish existence of solutions for doubly nonlinear evolution equations in reflexive Banach spaces by a De Giorgi–type energy-dissipation principle. It relaxes the problem to spaces of Banach-valued measures under a continuity equation, and employs a Von Neumann minimax approach to obtain a dual Hamilton–Jacobi formulation, together with a backward-boundedness argument to bound the dual and yield a zero minimum of the primal functional. The framework avoids time discretization and is robust to nonautonomous dissipation and time-dependent energies, with a structured three-step proof: convexification in measure spaces, duality reduction to a Hamilton–Jacobi constraint, and verification of a comparison-type principle ensuring zero dual-minimum. This provides a versatile existence theory for a broad class of rate-dependent dissipative systems and offers a flexible variational lens for analyzing doubly nonlinear flows under nonconvex energies and nonautonomous settings.

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.
Paper Structure (20 sections, 37 theorems, 224 equations)

This paper contains 20 sections, 37 theorems, 224 equations.

Table of Contents

  1. Notation and preliminaries
  2. Finite measures and narrow topology
  3. Radon-Nikodým spaces
  4. Fréchet and convex subdifferentials
  5. Fenchel conjugate
  6. Continuity equation and cylinder functions
  7. Abstract setting
  8. Reduction to a coercive and simpler setting
  9. Narrow topology for Banach-valued measures and action functional
  10. Narrow topology and compactness result
  11. The action functional
  12. Convexification in spaces of measures
  13. Minimax theorem and dual problem
  14. Non autonomous and state-dependent dissipation potential
  15. Proof of Theorem \ref{['thm:time']}
  16. ...and 5 more sections

Key Result

Theorem 1

Under the previous assumptions, for all initial datum $x_0\in \mathbb{R}^n$ there holds In particular, any minimizer solves eq:finitedim.

Theorems & Definitions (76)

  • Theorem 1
  • Theorem 2: Von Neumann
  • Remark 3: Time- and space-dependent dissipation
  • Remark 4: A Dynamic Programming viewpoint
  • Theorem 1.1: Prokhorov dellacherie1978probabilities
  • Lemma 1.2
  • proof
  • Definition 1.3: Cylinder functions
  • Definition 1.4
  • Lemma 1.5
  • ...and 66 more