Gamma convergence of a family of surface--director bending energies with small tilt

We prove a Gamma-convergence result for a family of bending energies defined on smooth surfaces in $\mathbb{R}^3$ equipped with a director field. The energies strongly penalize the deviation of the director from the surface unit normal and control the derivatives of the director. Such type of energies for example arise in a model for bilayer membranes introduced by Peletier and R\"oger [Arch. Ration. Mech. Anal. 193 (2009)]. Here we prove in three space dimensions in the vanishing-tilt limit a Gamma-liminf estimate with respect to a specific curvature energy. In order to obtain appropriate compactness and lower semi-continuity properties we use tools from geometric measure theory, in particular the concept of generalized Gauss graphs and curvature varifolds.Comment: 29 page

A Remark on the Anisotropic Outer Minkowski content

We study an anisotropic version of the outer Minkowski content of a closed set in Rn. In particular, we show that it exists on the same class of sets for which the classical outer Minkowski content coincides with the Hausdorff measure, and we give its explicit form.Comment: We corrected an error in the orignal manuscript, on p. 14 (the boundaries of the regularized sets are not necessarily C^{1,1}

Variational analysis of a mesoscale model for bilayer membranes

We present an asymptotic analysis of a mesoscale energy for bilayer membranes that has been introduced and analyzed in two space dimensions by the second and third author (Arch. Ration. Mech. Anal. 193, 2009). The energy is both non-local and non-convex. It combines a surface area and a Monge-Kantorovich-distance term, leading to a competition between preferences for maximally concentrated and maximally dispersed configurations. Here we extend key results of our previous analysis to the three dimensional case. First we prove a general lower estimate and formally identify a curvature energy in the zero-thickness limit. Secondly we construct a recovery sequence and prove a matching upper-bound estimate

Solution of the Kirchhoff-Plateau problem

The Kirchhoff-Plateau problem concerns the equilibrium shapes of a system in which a flexible filament in the form of a closed loop is spanned by a liquid film, with the filament being modeled as a Kirchhoff rod and the action of the spanning surface being solely due to surface tension. We establish the existence of an equilibrium shape that minimizes the total energy of the system under the physical constraint of non-interpenetration of matter, but allowing for points on the surface of the bounding loop to come into contact. In our treatment, the bounding loop retains a finite cross-sectional thickness and a nonvanishing volume, while the liquid film is represented by a set with finite two-dimensional Hausdorff measure. Moreover, the region where the liquid film touches the surface of the bounding loop is not prescribed a priori. Our mathematical results substantiate the physical relevance of the chosen model. Indeed, no matter how strong is the competition between surface tension and the elastic response of the filament, the system is always able to adjust to achieve a configuration that complies with the physical constraints encountered in experiments

Monotonicity formulae for smooth extremizers of integral functionals

A general monotonicity formula for smooth constrained local extremizers of firstorder integral functionals subject to non-holonomic constraints is established. The result is then applied to recover some known monotonicity formulae and to discover some new monotonicity formulae of potential value

The Plateau problem in the Calculus of Variations

This is a survey paper written for a course held for the Ph. D. program in Pure and Applied Mathematics at Politecnico di Torino during autumn 2018. The course has been dedicated to an overview of the main techniques for solving the Plateau problem, that is to find a surface with minimal area that spans a given boundary curve in the space. This problem dates back to the physical experiments of Plateau who tried to understand the possible configurations of soap films. From the mathematical point of view, the problem is very hard and a lot of possible formulations are available: perhaps still today none of these answers is the answer to the original formulation by Plateau. In this paper, first of all we will briefly introduce the problem showing that, at least in the smooth case, if the first variation of the area vanishes then the surface must have zero mean curvature. Then we will describe how the classical solution by Douglas and Radó works, and we will pass to modern formulations of the problem in the context of Geometric Measure Theory: sets of finite perimeter, currents, and minimal sets

Γ-Limits of convolution functionals

We compute the Γ-limit of a sequence of non-local integral functionals depending on a regularization of the gradient term by means of a convolution kernel. In particular, as Γ-limit, we obtain free discontinuity functionals with linear growth and with anisotropic surface energy density