Yann Bernard

Singularity removability at branch points for Willmore surfaces

Abstract: We consider a closed Willmore surface properly immersed in R with square-integrable second fundamental form, and with one point-singularity of finite arbitrary integer order. Using the “conservative” reformulation of the Willmore equation introduced in [Ri1], we show that, in an appropriate conformal parametrization, the gradient of the Gauss map of the immersion has bounded mean oscillations if the singularity has order one, and is bounded if the order is at least two. We develop around the singular point local asymptotic expansions for the immersion, its first and second derivatives, and for the mean curvature vector. Finally, we exhibit an explicit condition ensuring the removability of the point-singularity.

On the structure of minimizers of causal variational principles in the non-compact and equivariant settings

Abstract. We derive the Euler–Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.

Noether's theorem and the Willmore functional

Abstract Noether’s theorem and the invariances of the Willmore functional are used to derive conservation laws that are satisfied by the critical points of the Willmore energy subject to generic constraints. We recover in particular previous results independently obtained by R. Capovilla and J. Guven, and by T. Rivière. Several examples are considered in detail.

Analysis of constrained Willmore surfaces

ABSTRACT This paper investigates the regularity of constrained Willmore immersions into ℝm≥3 locally around both “regular” points and around branch points, where the immersive nature of the map degenerates. We develop local asymptotic expansions for the immersion and its first and second derivatives, given in terms of residues computed as circulation integrals. We deduce explicit “point removability” conditions ensuring that the immersion is smooth. Our results apply in particular to Willmore immersions and to parallel mean curvature immersions in any codimension.

Rigidity and stability of spheres in the Helfrich model

The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that idealised elastic phospholipid bilayers minimise H^{c_0} among all possible configurations. The functional integrates a spontaneous curvature parameter c_0 together with the mean curvature of the bilayer and constraints on area and volume, either through an inclusion of osmotic pressure difference and tensile stress or otherwise. Using the mathematical concept of embedded orientable surface to represent the configuration of the bilayer, one might expect to be able to adapt methods from differential geometry and the calculus of variations to perform a fine analysis of bilayer configurations in terms of the parameters that it depends upon. In this article we focus upon the case of spherical red blood cells with a view to better understanding spherocytes and spherocytosis. We provide a complete classification of spherical solutions in terms of the parameters in the Helfrich model. We additionally present some further analysis on the rigidity and stability of spherocytes.

Diffusive limits for the Knudsen gas in a thin channel with accommodation on the boundary

We consider a model for a free molecular flow in a thin channel bounded by two parallel plates on which Maxwellian boundary conditions with (fixed) accommodation and a generic scattering kernel apply. Using functional analytic tools, we show that as the width of the channel vanishes, and on a suitable temporal scale, the evolution of the density is described by a diffusion problem. We distinguish two classes of temporal scalings (normal and anomalous) and we show that an infinitesimal amount of grazing collisions with the walls of the channel is responsible for the anomalous diffusion. The method employed is adapted from the original work of F. Golse in Asymptot. Anal. 17 (1998), 1-12.