We consider general surface energies, which are weighted integrals over a closed surface with a weight function depending on the position, the unit normal and the mean curvature of the surface. Energies of this form have applications in many areas, such as materials science, biology and image processing. Often one is interested in finding a surface that minimizes such an energy, which entails finding its first variation with respect to perturbations of the surface. We present a concise derivation...

Dans ce travail: (1) on caractérise l’espace $C_G$ des fonctionnelles invariantes par un groupe compact G opérant linéairement et continûment sur un espace vectoriel topologique localement convexe séparé et séquentiellement complet E plus précisément, on montre que $C_G$ est le dual topologique du sous-espace $E_G$ des vecteurs de E qui sont G-invariants. (2) On étudie les courants basiques sur une variété feuilletée (V,ℱ). On obtient alors, dans le cas où le feuilletage est associé à une action localement...

The distributional $k$-dimensional Jacobian of a map $u$ in the Sobolev space $W^{1,k-1}$ which takes values in the sphere $S^{k-1}$ can be viewed as the boundary of a rectifiable current of codimension $k$ carried by (part of) the singularity of $u$ which is topologically relevant. The main purpose of this paper is to investigate the range of the Jacobian operator; in particular, we show that any boundary $M$ of codimension $k$ can be realized as Jacobian of a Sobolev map valued in $S^{k-1}$. In case $M$ is polyhedral, the map we construct...

We establish some new results about the Γ-limit, with respect to the L1-topology, of two different (but related) phase-field approximations $\{{ℰ}_{\}},\{{\widetilde{ℰ}}_{\}}$ ℰ ε ε ,   x10ff65; ℰ ε ε of the so-called Euler’s Elastica Bending Energy for curves in the plane. In particular we characterize theΓ-limit as ε → 0 of ℰε, and show that in general the Γ-limits of ℰεand $\widetilde{ℰ}$ x10ff65; ℰ ε do not coincide on indicator functions of sets with non-smooth boundary. More precisely we show that the domain of theΓ-limit of $\widetilde{ℰ}$ x10ff65;...

We introduce the space $GBD$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $GBD$, which leads to a compactness result for the space $GSBD$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation of some variational...

Using a calibration method we prove that, if $\Gamma \subset \Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\Gamma$ and regular outside, then the function $u_{\beta }$ which solves$$\left\{\begin{array}{cc}\Delta u_{\beta }=\beta (u_{\beta }-g)\hfill & \text{in}\Omega \setminus \Gamma \hfill \\ {\partial }_{\nu }{u}_{\beta }=0\hfill & \text{on}\partial \Omega \cup \Gamma \hfill \end{array}\right.$$is in turn discontinuous along $\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional$${\int }_{\Omega \setminus S_u}{\left|\nabla u\right|}^{2}dx+{\mathscr{H}}^{n-1}\left(S_u\right)+\beta {\int }_{\Omega \setminus S_u}{(u-g)}^{2}dx,$$over $SBV\left(\Omega \right)$, for $\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

The problem of distributing two conducting materials with a prescribed volume ratio in a ball so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions is considered in two and three dimensions. The gap ε between the two conductivities is assumed to be small (low contrast regime). The main result of the paper is to show, using asymptotic expansions with respect to ε and to small geometric perturbations of the optimal shape, that the global minimum of the first eigenvalue...

We consider a Canham − Helfrich − type variational problem defined over closed surfaces enclosing a fixed volume and having fixed surface area. The problem models the shape of multiphase biomembranes. It consists of minimizing the sum of the Canham − Helfrich energy, in which the bending rigidities and spontaneous curvatures are now phase-dependent, and a line tension penalization for the phase interfaces. By restricting attention to axisymmetric surfaces and phase distributions, we extend our previous...