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Variational principles and apllications to symmetric PDEs

Javier Falco, Daniel Isert

Abstract

In this paper, we explore various equivalences of Ekeland's variational principle within the framework of group-invariant mappings. We introduce and analyze several key theorems, including the Drop theorem, the Petal theorem, Caristi-Kirk fxed-point theorem, and Takahashi's theorem, all of them within this context. Moreover, we extend the classical Drop theorem and Petal theorem to a more generalized setting. We also demonstrate the practical signifcance of these findings through numerous applications to diverse areas of mathematics. In particular, in the context of partial differential equations, we explore their implications on the solution of the Plateau problem, and in control theory. We also extend the classical Pontyargin maximum principle.

Variational principles and apllications to symmetric PDEs

Abstract

In this paper, we explore various equivalences of Ekeland's variational principle within the framework of group-invariant mappings. We introduce and analyze several key theorems, including the Drop theorem, the Petal theorem, Caristi-Kirk fxed-point theorem, and Takahashi's theorem, all of them within this context. Moreover, we extend the classical Drop theorem and Petal theorem to a more generalized setting. We also demonstrate the practical signifcance of these findings through numerous applications to diverse areas of mathematics. In particular, in the context of partial differential equations, we explore their implications on the solution of the Plateau problem, and in control theory. We also extend the classical Pontyargin maximum principle.
Paper Structure (10 sections, 19 theorems, 127 equations, 2 figures)

This paper contains 10 sections, 19 theorems, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivalence of the Drop theorem, Petal theorem, and Ekeland's variational principle
  4. Caristi-Kirk, Ekeland and Takahasi's theorem
  5. Applications of the Ekeland variational principle
  6. Drop and generalized Drop theorem
  7. Partial differential equations
  8. Plateau problem
  9. General patial differential equations
  10. Control theory

Key Result

Proposition 2

Let $(M, d)$ be a metric space and $G \subseteq \mathcal{L}(M)$ be a topological group of isometries acting on $M$. Let $\gamma > 0$, then

Figures (2)

  • Figure 1: Flower theorem statement.
  • Figure 2: Generalized Drop theorem statement.

Theorems & Definitions (38)

  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 5
  • Corollary 6
  • Example 7
  • Example 8
  • Theorem 11
  • ...and 28 more