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The Rabinowitz minimal periodic solution conjecture on partially convex reversible Hamiltonian systems and brake subharmonics

Yuting Zhou

TL;DR

This work advances the Rabinowitz minimal periodic solution conjecture for partially convex reversible Hamiltonian systems by developing a generalized mountain-pass essential point framework and a novel dual action functional that accommodates weaker regularity. It proves the existence of $T$-periodic brake solutions with minimal period $T$ for autonomous systems under a semidefinite $H_{pp}$ condition (or $n=1$), and it extends these results to brake subharmonics in nonautonomous reversible settings. A central novelty is the synthesis of a refined Morse index analysis with Maslov-type indices, including new iteration inequalities and a relative Morse index, enabling precise control of periodicity and the number of distinct subharmonics. The combination of variational methods, duality, and index theory yields both minimal-period results and a rich subharmonic structure, with significant implications for the study of symmetric Hamiltonian dynamics and brake orbits.

Abstract

Under weaker regularity and compactness assumptions, we find the mountain-pass essential point, which is a novel extension of the classical Ambrosetti-Rabinowitz mountain pass theorem. We study the reversible superquadratic autonomous Hamiltonian systems whose Hamiltonian $H(p,q)$ is strictly convex in the position $q\in\mathbf{R}^n$ and prove that for every $T>0$, the system has a $T$-periodic brake solution $\bar x$ with minimal period $T$, provided the Hessian $H_{pp}(\bar x(t))\in\mathbf{R}^{n\times n}$ is semi-positive definite for $t\in\mathbf{R}$ or $n=1$. For brake subharmonics of general reversible nonautonomous Hamiltonian systems, we also get some new results.

The Rabinowitz minimal periodic solution conjecture on partially convex reversible Hamiltonian systems and brake subharmonics

TL;DR

This work advances the Rabinowitz minimal periodic solution conjecture for partially convex reversible Hamiltonian systems by developing a generalized mountain-pass essential point framework and a novel dual action functional that accommodates weaker regularity. It proves the existence of -periodic brake solutions with minimal period for autonomous systems under a semidefinite condition (or ), and it extends these results to brake subharmonics in nonautonomous reversible settings. A central novelty is the synthesis of a refined Morse index analysis with Maslov-type indices, including new iteration inequalities and a relative Morse index, enabling precise control of periodicity and the number of distinct subharmonics. The combination of variational methods, duality, and index theory yields both minimal-period results and a rich subharmonic structure, with significant implications for the study of symmetric Hamiltonian dynamics and brake orbits.

Abstract

Under weaker regularity and compactness assumptions, we find the mountain-pass essential point, which is a novel extension of the classical Ambrosetti-Rabinowitz mountain pass theorem. We study the reversible superquadratic autonomous Hamiltonian systems whose Hamiltonian is strictly convex in the position and prove that for every , the system has a -periodic brake solution with minimal period , provided the Hessian is semi-positive definite for or . For brake subharmonics of general reversible nonautonomous Hamiltonian systems, we also get some new results.

Paper Structure

This paper contains 25 sections, 64 theorems, 385 equations.

Table of Contents

  1. Introduction and Main Results
  2. Extension of the mountain-pass essential point
  3. Minimal periodic solution problem for autonomous Hamiltonian systems
  4. Main results
  5. Methods and new observations
  6. Multiple subharmonic solutions for nonautonomous Hamiltonian systems
  7. Study of the critical point
  8. Existence of the mountain-pass essential point
  9. Application of Theorem \ref{['t:critical-point-mp']} to partially convex reversible Hamiltonian systems
  10. Morse index and the geometrical structure near the critical point
  11. Iteration theory of Maslov-type indices: old and new
  12. Iteration formulae and inequalities of the Maslov-type index theory
  13. Proof of Theorem \ref{['t:super-quadratic-1']}
  14. The Morse index, the Maslov-type indices and the relative Morse index
  15. Relation between the Morse and the Maslov-type index of a solution
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $f\colon E\to \mathbf{R}$ be a continuous and Gâteaux-differentiable function on a Banach space $E$ such that $f'\colon E\to E^*$ is continuous from the norm topology of $E$ to the weak$^*$-topology of $E^*$. Take two distinct points $(u_0,u_1)$ in $E$, and define Assume $f$ satisfies condition (C) at the level $d$ (see Definition d:conditionC). If then there is a $\bar{u}\in \mathop{\mathrm

Theorems & Definitions (132)

  • Definition 1.1: cf. Hofer85
  • Theorem 1.2: The Generalized Mountain Pass Theorem
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • ...and 122 more