Lower semicontinuity for the Helfrich problem
- PDF / 2,566,239 Bytes
- 29 Pages / 439.37 x 666.142 pts Page_size
- 94 Downloads / 203 Views
Lower semicontinuity for the Helfrich problem Sascha Eichmann1 Received: 3 September 2019 / Accepted: 20 May 2020 © The Author(s) 2020
Abstract We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of ℝ3. Keywords Closed immersions · Canham–Helfrich energy · Lower semicontinuity · Variational problem · Oriented varifolds Mathematics Subject Classification 26B15 · 35J35 · 49J45 · 49Q20 · 58J90
1 Introduction This article deals with minimising the Helfrich energy for closed oriented smooth connected two-dimensional immersions f ∶ Σ → ℝ3 , which is defined as
WH0 (f ) ∶=
∫Σ
(H̄ − H0 )2 d𝜇g .
(1.1)
Here H̄ is the scalar mean curvature, i.e. the sum of the principal curvatures with respect to a choosen unit normal of f. We call 𝜇g the area measure on Σ induced by f and the euclidean metric of ℝ3 . Furthermore, H0 ∈ ℝ is called spontaneous curvature. This energy was introduced by Helfrich [20] and Canham [3] to model the shape of blood cells. Hence it is
The author thanks Prof. Reiner Schätzle for discussing the Helfrich energy and providing insight into geometric measure theory. * Sascha Eichmann [email protected]‑tuebingen.de 1
Mathematisch‑Naturwissenschaftliche Fakultät, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
13
Vol.:(0123456789)
Annals of Global Analysis and Geometry
called the Canham–Helfrich or short Helfrich energy. We recover the Willmore energy by setting H0 = 0 and multiplying by 14 , i.e.
WWill (f ) ∶=
1 ̄ 2 dA = 1 W0 (f ). |H| 4 ∫Σ 4
(1.2)
The Willmore energy goes back to Thomsen [40]. He denoted critical points of the Willmore energy as conformal minimal surfaces. Willmore later revived the mathematical discussion in [41]. Please note, that for the Willmore energy an orientation is not needed, contrary to the Helfrich energy. Hence we need a fixated normal. We assume Σ to have a continuous orientation 𝜏 , given as a 2-form. Then we set 𝜈f ∶=∗ (df (𝜏)) ∈ 𝜕B1 (0) ⊂ ℝ3 as the unit normal of f. Here ∗ denotes the Hodge-∗-Operator. Furthermore, we like to prescribe the area and enclosed volume of f. Therefore we set
Area(f ) ∶=
∫Σ
d𝜇g ,
Vol(f ) ∶=
1 f (x) ⋅ 𝜈f (x) d𝜇g (x). 3 ∫Σ
(1.3)
Please note that, if f is an embedding and 𝜈f the outer normal, Vol(f) would be the volume of the set enclosed by f. In the general case, Vol(f) may become negat
Data Loading...
