Li–Yau inequalities for the Helfrich functional and applications

Abstract

We prove a general Li–Yau inequality for the Helfrich functional where the spontaneous curvature enters with a singular volume type integral. In the physically relevant cases, this term can be converted into an explicit energy threshold that guarantees embeddedness. We then apply our result to the spherical case of the variational Canham–Helfrich model. If the infimum energy is not too large, we show existence of smoothly embedded minimizers. Previously, existence of minimizers was only known in the classes of immersed bubble trees or curvature varifolds.

Curvature-dependent Eulerian interfaces in elastic solids

“…To incorporate the penalization of large interface curvatures, we describe the interface in terms of a curvature varifold, a measure-theoretic generalization of classical surfaces with a notion of curvature and with good compactness properties [16][17][18]. Mathematical models involving varifolds have been used to describe bending-resistant interfaces in a wide range of applications, for instance in the modelling of cracks [19][20][21], biological membranes [22][23][24] or anisotropic phase transitions [25].…”

Bifurcation of elastic curves with modulated stiffness

“…To incorporate the penalization of large interface curvatures, we describe the interface in terms of a curvature varifold, a measure-theoretic generalization of classical surfaces with a notion of curvature and with good compactness properties [20,23,29]. Varifolds have been used to describe bending-resistant interfaces in a wide range of applications, for instance in the modeling of cracks [15,16,22], biological membranes [10,13,26], or anisotropic phase transitions [25].…”