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Nonsmooth Analysis of Doubly Nonlinear Second-Order Evolution Equations with Non-Convex Energy Functionals

Aras Bacho

Abstract

We investigate the existence of strong solutions to a general class of doubly multivalued and nonlinear evolution equations of second order. The multivalued operators are generated by the subdifferential of nonsmooth potentials that live in different spaces, $U$ and $V$, where in general $U \nsubseteq V$ and $V \nsubseteq U$. The proof is based on the regularization of the dissipation potential using the generalized Moreau--Yosida regularization and a semi-implicit time-discretization scheme, which demonstrates the existence of strong solutions to the regularized problem. The existence of solutions to the original problem is then shown by letting the regularization parameter converge to zero. Furthermore, we establish an energy-dissipation inequality for the solution. We conclude with applications of this abstract theory.

Nonsmooth Analysis of Doubly Nonlinear Second-Order Evolution Equations with Non-Convex Energy Functionals

Abstract

We investigate the existence of strong solutions to a general class of doubly multivalued and nonlinear evolution equations of second order. The multivalued operators are generated by the subdifferential of nonsmooth potentials that live in different spaces, and , where in general and . The proof is based on the regularization of the dissipation potential using the generalized Moreau--Yosida regularization and a semi-implicit time-discretization scheme, which demonstrates the existence of strong solutions to the regularized problem. The existence of solutions to the original problem is then shown by letting the regularization parameter converge to zero. Furthermore, we establish an energy-dissipation inequality for the solution. We conclude with applications of this abstract theory.

Paper Structure

This paper contains 22 sections, 9 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Literature review
  4. Organization of the paper
  5. Topological assumptions and preliminaries
  6. Function space setting
  7. Facts from convex analysis
  8. Main result
  9. Assumptions on the functionals and operators
  10. Discussion of the assumptions
  11. Statement of the main result
  12. Proof of the main result
  13. Regularized problem
  14. Variational approximation scheme
  15. Discrete Energy-Dissipation inequality and a priori estimates
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $F_1:X\rightarrow (-\infty,+\infty]$ and $F_2:X\rightarrow (-\infty,+\infty]$ be subdifferentiable and Fréchet differentiable at $u\in \operatorname{dom}(\partial F_1)\cap \operatorname{dom}(D F_2)\neq \emptyset$, respectively. Then, there holds where $D F_2$ denotes the Fréchet derivative of $F_2$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Variational sum rule
  • proof
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 13 more