This paper deals with the analysis of a class of doubly nonlinear evolution equations in the framework of a general metric space. We propose for such equations a suitable metric formulation (which in fact extends the notion of Curve of Maximal Slopefor gradient flows in metric spaces, see [5]), and prove the existence of solutions for the related Cauchy problem by means of an approximation scheme by time discretization. Then, we apply our results to obtain the existence of solutions to abstract...

This paper gives a new proof of the fact that a $k$-dimensional normal current $T$ in ${ℝ}^m$ is integer multiplicity rectifiable if and only if for every projection $P$ onto a $k$-dimensional subspace, almost every slice of $T$ by $P$ is $0$-dimensional integer multiplicity rectifiable, in other words, a sum of Dirac masses with integer weights. This is a special case of the Rectifiable Slices Theorem, which was first proved a few years ago by B. White.

We describe a new link between Perelman’s monotonicity formula for the reduced volume and ideas from optimal transport theory.


We consider, in an open subset Ω of ${ℝ}^N$, energies depending on the perimeter of a subset $E\subset \Omega$ (or some equivalent surface integral) and on a function u which is defined only on $\Omega \setminus E$. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application,...

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller. Let $H=\left(\begin{array}{cc}\sigma & 0\\ 0& {\sigma }^{-1}\end{array}\right)$ for $\sigma \>0$. Let $0\<{\zeta }_1\<1\<{\zeta }_2\<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in W^{2,1}\left(Q_1\left(0\right)\right)$ be a ${\mathrm{C}}^1$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)\<{\zeta }_2$, $\mathrm{Lip}\left(u^{-1}\right)\<{\zeta }_1^{-1}$. There exists positive constants ${𝔠}_1\<1$ and ${𝔠}_2\>1$ depending only on $\sigma$, ${\zeta }_1$, ${\zeta }_2$ such that if $ϵ\in \left(0,{𝔠}_1\right)$ and $u$ satisfies the...

In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let $H=\left(\begin{array}{ccc}\sigma & 00& {\sigma }^{-1}\end{array}\right)$ for $\sigma >0$. Let $0<{\zeta }_1<1<{\zeta }_2<\infty$. Let $K:=SO\left(2\right)\cup SO\left(2\right)H$. Let $u\in W^{2,1}\left(Q_1\left(0\right)\right)$ be a $$ invertible bilipschitz function with $\mathrm{Lip}\left(u\right)<{\zeta }_2$, $\mathrm{Lip}\left(u^{-1}\right)<{\zeta }_1^{-1}$. 
There exists positive constants ${𝔠}_1<1$ and ${𝔠}_2>1$ depending only on σ, ${\zeta }_1$, ${\zeta }_2$ such that if $ϵ\in \left(0,{𝔠}_1\right)$ and u satisfies...

We propose a variational model to describe the optimal distributions of residents and services in an urban area. The functional to be minimized involves an overall transportation cost taking into account congestion effects and two aditional terms which penalize concentration of residents and dispersion of services. We study regularity properties of the minimizers and treat in details some examples.

We propose a variational model to describe the optimal distributions of residents and services in an urban area. The functional to be minimized involves an overall transportation cost taking into account congestion effects and two aditional terms which penalize concentration of residents and dispersion of services. We study regularity properties of the minimizers and treat in details some examples.

We discuss a variational problem defined on couples of functions that are constrained to take values into the 2-dimensional unit sphere. The energy functional contains, besides standard Dirichlet energies, a non-local interaction term that depends on the distance between the gradients of the two functions. Different gradients are preferred or penalized in this model, in dependence of the sign of the interaction term. In this paper we study the lower semicontinuity and the coercivity of the energy...

Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms...

Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms...

In the sub-Riemannian framework, we give geometric necessary and sufficient conditions for the existence of abnormal extremals of the Maximum Principle. We give relations between abnormality, $C^1$-rigidity and length minimizing. In particular, in the case of three dimensional manifolds we show that, if there exist abnormal extremals, generically, they are locally length minimizing and in the case of four dimensional manifolds we exhibit abnormal extremals which are not $C^1$-rigid and which can be minimizing...