Polar tangential angles and free elasticae

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 ∈ (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 ∈ (2, ∞) there emerge uncountably many local minimizers with diverging energy.

Willmore obstacle problems under Dirichlet boundary conditions

“…For symmetric cone obstacles ψ, i.e., ψ(x) = ψ(1 − x) for all x ∈ [0, 1] and ψ is affine on (0, 1/2), such that ψ(0) = ψ(1) < 0 and ψ(1/2) > 0, the question was completely solved as follows: (i) if ψ(1/2) < 2/c 0 , then there exists a unique minimiser of W in K sym (ψ); (ii) if ψ(1/2) ≥ 2/c 0 , then there is no minimiser of W not only in K sym (ψ) but also in K(ψ), where We note that the existence of a minimiser in the first case follows from Dall'Acqua-Deckelnick [1]. Its uniqueness was independently proved by Miura [9] and Yoshizawa [16]. The non-existence of minimisers in the case ψ(1/2) > 2/c 0 was proved by Müller [10] and in the critical case ψ(1/2) = 2/c 0 by Yoshizawa [16].…”

“…More precisely, we consider minimisation problems for the Willmore functional among curves or surfaces of revolution with Dirichlet boundary conditions under a unilateral constraint. Recently the minimisation problem for the one-dimensional Willmore or elastica functional among graphs of functions with Navier boundary conditions under a unilateral constraint has been intensively studied ( [1,9,10,16]): For a given obstacle function ψ : [0, 1] → R, find a function u : [0, 1] → R such that u attains (1.1) inf…”

The Elastic Flow with Obstacles: Small Obstacle Results

“…Existence (and nonexistence) of minimizers of E in C ψ has been studied in [7] and [18], minimization with slightly different frameworks has also been examined in [15], [16], [17] and [8].…”

“…The articles [7], [18] and [17] reveal that under certain smallness conditions on ψ minimizers do exist whereas they do not exist in general if the obstacle is too large.…”

“…One question that could be asked is how many critical points exist. A partial answer is given in [19,Corollary 5.22] and [17], where it is shown that there exist no critical points above an obstacle of a certain height. This is also why our convergence results may only hold true for small obstacles.…”