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Obstacle problems for the elastic flow and related topics

Kensuke Yoshizawa

TL;DR

This work analyzes an obstacle problem for the length-penalized elastic flow in the plane, formulating a graphical evolution constrained by an obstacle and fixed endpoints. It develops a local-in-time weak solution theory via a minimizing movements scheme for the energy $\mathcal{E}_\lambda$ and proves that, under smoothness, the weak solution recovers the classical elastic flow with the energy-dissipation law $\frac{d}{dt}\mathcal{E}_\lambda[\gamma_u(t)]= -\int|\partial_t^\perp\gamma_u|^2\,ds$. The paper also discusses the potential loss of regularity, the link to rectangular-elastica configurations, and situates the results within the broader literature on obstacle problems for bending energy, including dynamics in different metric settings. Overall, it provides a rigorous bridge between variational weak formulations and classical geometric evolutions under obstacle constraints, with implications for regularity and energy-dissipation analysis in constrained elastic flows.

Abstract

In this note, we study an obstacle problem for the elastic flow. We prove the local-in-time existence of weak solutions and discuss their relation to classical solutions when additional regularity is obtained. Related results concerning obstacle problems for the bending energy are also collected.

Obstacle problems for the elastic flow and related topics

TL;DR

This work analyzes an obstacle problem for the length-penalized elastic flow in the plane, formulating a graphical evolution constrained by an obstacle and fixed endpoints. It develops a local-in-time weak solution theory via a minimizing movements scheme for the energy and proves that, under smoothness, the weak solution recovers the classical elastic flow with the energy-dissipation law . The paper also discusses the potential loss of regularity, the link to rectangular-elastica configurations, and situates the results within the broader literature on obstacle problems for bending energy, including dynamics in different metric settings. Overall, it provides a rigorous bridge between variational weak formulations and classical geometric evolutions under obstacle constraints, with implications for regularity and energy-dissipation analysis in constrained elastic flows.

Abstract

In this note, we study an obstacle problem for the elastic flow. We prove the local-in-time existence of weak solutions and discuss their relation to classical solutions when additional regularity is obtained. Related results concerning obstacle problems for the bending energy are also collected.
Paper Structure (11 sections, 12 theorems, 153 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 153 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of related work
  3. From weak to classical solutions
  4. Remarks on the weak solution
  5. Consistency under smoothness assumptions
  6. Loss of regularity
  7. Existence of a local-in-time solution
  8. Construction of scheme
  9. Regularity
  10. Convergence
  11. Elliptic integrals and functions

Key Result

Theorem 1

Let $\lambda\geq0$ and suppose that $\psi \in C(\bar{I})$ satisfies eq:psi_condition. Then, for each $u_0 \in K$ there exists $T=T(u_0)>0$ such that eq:P possesses a weak solution $u$, which satisfies for any $p\in[2,\infty)$.

Figures (1)

  • Figure 1: A rectangular elastica $\Gamma_{\rm rect}$ (left), the unique symmetric solution $u$ of \ref{['eq:VI_station']} with a symmetric cone obstacle whose height is $h_*/2$. The third weak derivative of $u$ is discontinuous at $x=1/2$ (center). A part of $\Gamma_{\rm rect}$ which coincides with $u|_{[0,\frac{1}{2}]}$ in the center, after a suitable rotation and dilation (right).

Theorems & Definitions (26)

  • Definition 1: Weak solution
  • Theorem 1: Existence of local-in-time solution
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Lemma 1
  • proof
  • Proposition 2
  • Definition 2
  • ...and 16 more