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Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application

Thomas Ruf

Abstract

This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for generalized gradient flows, overcoming challenges such as non-coercive and infinity-valued dissipation potentials and non-monotone subdifferential operators on non-reflexive Banach spaces. The second part focuses on the study of Banach-valued Orlicz spaces, a flexible class of Banach spaces for quantifying the growth of nonlinear functions. Besides improving many known results by imposing minimal assumptions, we extend the theory by handling infinity-valued Orlicz integrands and arbitrary Banach-values in the duality theory. The combination of these results offers a powerful tool for analyzing differential equations involving functions of arbitrary growth rates and leads to a significant improvement over previous results, demonstrated through the existence of weak solutions for a doubly nonlinear initial-boundary value problem of Allen-Cahn-Gurtin type.

Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application

Abstract

This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for generalized gradient flows, overcoming challenges such as non-coercive and infinity-valued dissipation potentials and non-monotone subdifferential operators on non-reflexive Banach spaces. The second part focuses on the study of Banach-valued Orlicz spaces, a flexible class of Banach spaces for quantifying the growth of nonlinear functions. Besides improving many known results by imposing minimal assumptions, we extend the theory by handling infinity-valued Orlicz integrands and arbitrary Banach-values in the duality theory. The combination of these results offers a powerful tool for analyzing differential equations involving functions of arbitrary growth rates and leads to a significant improvement over previous results, demonstrated through the existence of weak solutions for a doubly nonlinear initial-boundary value problem of Allen-Cahn-Gurtin type.
Paper Structure (75 sections, 114 theorems, 676 equations)

This paper contains 75 sections, 114 theorems, 676 equations.

Table of Contents

  1. Introduction
  2. Subject matters
  3. Open and follow-up questions.
  4. Structure of the thesis
  5. Generalized Gradient Flows
  6. Generalized Orlicz Spaces
  7. Appendix
  8. Notation and conventions
  9. Generalized gradient flows
  10. Setting and assumptions
  11. The spaces
  12. Remark and examples of setting.
  13. The dissipation potentials
  14. Remark on assumptions.
  15. The energy functional
  16. ...and 60 more sections

Key Result

Proposition 2.1

If $\mu$ is a measure and $f_n, f$ are non-negative measurable functions such that $f \le \liminf_n f_n$ a.e. and $\lim_n \| f_n \|_{L_1} = \| f \|_{L_1} < \infty$, then $\lim_n \| f - f_n \|_{L_1} = 0$.

Theorems & Definitions (243)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 233 more