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...

A gradient estimate for solutions to the minimal surface equation can be proved by Partial Differential Equations methods, as in [2]. In such a case, the oscillation of the solution controls its gradient. In the article presented here, the estimate is derived from the Harnack type inequality established in [1]. In our case, the gradient is controlled by the area of the graph of the solution or by the integral of it. These new results are similar to the one announced by Ennio De Giorgi in [3].

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in ${ℝ}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step...

We introduce a new transport distance between probability measures on ${ℝ}^d$ that is built from a Lévy jump kernel. It is defined via a non-local variant of the Benamou–Brenier formula. We study geometric and topological properties of this distance, in particular we prove existence of geodesics. For translation invariant jump kernels we identify the semigroup generated by the associated non-local operator as the gradient flow of the relative entropy w.r.t. the new distance and show that the entropy is...

A unified framework for analyzing the existence of ground states in wide classes of elastic complex bodies is presented here. The approach makes use of classical semicontinuity results, Sobolev mappings and cartesian currents. Weak diffeomorphisms are used to represent macroscopic deformations. Sobolev maps and cartesian currents describe the inner substructure of the material elements. Balance equations for irregular minimizers are derived. A contribution to the debate about the role of the balance...