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.

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...