Variational stabilization of degenerate p-elasticae

TL;DR

This work demonstrates a novel stabilization phenomenon for pinned planar $p$-elasticae under degenerate diffusion ($p>2$), showing the existence of uncountably many local minimizers with unbounded energy. The authors introduce hooked boundary conditions to enable a geometric relaxation, classify hooked $p$-elasticae (wave-like and flat-core), and establish global minimizers with explicit energy formulas via $p$-elliptic integrals. They further prove stability of alternating flat-core pinned $p$-elasticae by decomposing into hooked pieces and applying a global-minimization bound, revealing how degeneracy can generate robust, high-energy local minimizers. Collectively, the results illuminate a distinct variational mechanism in geometric elasticity induced by degenerate diffusion and bear on the broader study of nonlinear diffusion and higher-order energy landscapes.

Abstract

A new stabilization phenomenon induced by degenerate diffusion is discovered in the context of pinned planar $p$-elasticae. It was known that in the non-degenerate regime $p\in(1,2]$, including the classical case of Euler's elastica, there are no local minimizers other than unique global minimizers. Here we prove that, in stark contrast, in the degenerate regime $p\in(2,\infty)$ there emerge uncountably many local minimizers with diverging energy.

Figures (3)

• Figure 1: Examples of flat-core pinned $p$-elasticae. (i), (ii) Unstable MY_Crelle. (iii) Stable (alternating, Theorem \ref{['thm:alternating_stability']}). (iv) Open (quasi-alternating, Problem \ref{['prob:quasi-open']}).
• Figure 2: The profile of the loop $\gamma^+_b$.
• Figure 3: Decomposition of an alternating flat-core $p$-elastica.

Theorems & Definitions (42)

• Theorem 1.1: MY_Crelle
• Theorem 1.2
• Theorem 1.3: Theorem \ref{['thm:alternating_detailed']}
• Definition 2.1
• Lemma 2.2: nabe14*Lemma 2
• Definition 2.3
• Proposition 2.4: MY_AMPA
• Proposition 2.5: MY_AMPA*Theorem 1.7 and Lemma 4.3
• Proposition 2.6: MY_AMPA*Theorems 1.2, 1.3
• Remark 2.7