Boundary value problems for Willmore curves in $$\mathbb {R}^2$$ R 2

Abstract

In this paper the Navier problem and the Dirichlet problem for Willmore curves in R 2 are solved.

Explicit formulas, symmetry and symmetry breaking for Willmore surfaces of revolution

“…Since we will not make use of this ODE we omit the proof. (b) A similar result may be shown for elastic curves in the Euclidean plane that have recently been studied in [6,19]. Such curvesγ, now parametrized by Euclidean arclength with curvatureκ, satisfyκẐ ′′ − 2κ ′Ẑ ′ = const whereẐ(s) :…”

“…Notice that κ = κ 0 cn 2 (rs, p) is a typo in [15] since the curvature function is basically a dn-function (cf. (19)), as we will see later. Following [15] (not necessarily closed) elastic curves generated by such curvature functions will be called orbitlike.…”

“…Proposition 3. Let κ be given by (19). Then {W 1 , W 2 } is a fundamental system of the ODE W ′′ + 2µκ −2 W = 0 and the following identities hold on R:…”

Periodic Solutions to a Cahn–Hilliard–Willmore Equation in the Plane

“…The terms ε 3 Lφ (εs)η(t) and ε 2 φ (εs) 2η (t), involving the Fourier truncation φ might seem more delicate. However, being L of second order (see (29)), using (24) and recalling that φ C 4,α ≤ c ε, one has that the x 2 -derivative of both these terms is of order ε 4 .…”

“…This equation can be explicitly solved using special functions, and then integrated to produce the above Willmore curves γ T . Indeed, every non-affine complete planar Willmore curve coincides, up to an affine transformation with the curve γ T , see [29]. Apart from producing a first non-compact profile of this type for the equation, our aim is to explore the relation between one-dimensionality of solutions and their limit properties.…”