Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation.
Regularity of solutions fo a degenerate elliptic variational problem.
Metric space valued functions of bounded variation
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems
n this note we describe some recent developments of the theory of flows associated to vector fields with a low regularity with respect to the spatial variables, for instance with a Sobolev or BV regularity. After the illustration of some applica- tions of this theory to conservation laws and PDE's in fluid dynamics, we give an axiomatic presentation of the problem, based on a probabilistic approach inspired by the work of L.C. Young. In the final part we discuss very recent results on the regularity...
Problema di trasporto e equazione di Cauchy per campi vettoriali a variazione limitata
In questa conferenza descrivo alcuni recenti sviluppi relativi al problema dell'unicità per l'equazione differenziale ordinaria e per l'equazione di continuità per campi vettoriali debolmente differenziabili. Descrivo infine un'applicazione di questi risultati a un sistema di leggi di conservazione.
On some recent developments of the theory of sets of finite perimeter
In this paper we describe some recent progress on the theory of sets of finite perimeter, currents, and rectifiability in metric spaces. We discuss the relation between intrinsic and extrinsic theories for rectifiability
Su alcune proprietà delle funzioni convesse
In questo lavoro riassumiamo alcuni risultati di una ricerca riguardante le singolarità (punti di non differenziabilità) delle funzioni convesse. Questa ricerca copre vari aspetti, che vanno dalla stima della dimensione di Hausdorff di certi tipi di singolarità fino allo studio della loro propagazione. Studiamo anche problemi di semicontinuità e rilassamento collegati all'area del grafico del gradiente di una funzione convessa e l'esistenza dei determinanti, in senso debole, dei minori della matrice...
Nuovi risultati sulla semicontinuità inferiore di certi funzionali integrali
Given an open subset of and a Borel function , conditions on are given which assure the lower semicontinuity of the functional with respect to different topologies.
Transport equation and Cauchy problem for vector fields and applications
Nuovi risultati sulla semicontinuità inferiore di certi funzionali integrali
Given an open subset of and a Borel function , conditions on are given which assure the lower semicontinuity of the functional with respect to different topologies.
Gradient Flows in Metric Spaces and in the Spaces of Probability Measures, and Applications to Fokker-Planck Equations with Respect to Log-Concave Measures
A survey on the main results of the theory of gradient flows in metric spaces and in the Wasserstein space of probability measures obtained in [3] and [4], is presented.
Compactness of Special Functions of Bounded Higher Variation
Given an open set Ω ⊂ Rm and n > 1, we introduce the new spaces GBnV(Ω) of Generalized functions of bounded higher variation and GSBnV(Ω) of Generalized special functions of bounded higher variation that generalize, respectively, the space BnV introduced by Jerrard and Soner in [43] and the corresponding SBnV space studied by De Lellis in [24]. In this class of spaces, which allow as in [43] the description of singularities of codimension n, the distributional jacobian Ju need not have finite...
Partial regularity of free discontinuity sets, I
Surface measures and convergence of the Ornstein-Uhlenbeck semigroup in Wiener spaces
We study points of density of sets of finite perimeter in infinite-dimensional Gaussian spaces and prove that, as in the finite-dimensional theory, the surface measure is concentrated on this class of points. Here density is formulated in terms of the pointwise behaviour of the Ornstein-Uhlembeck semigroup.
On the Regularity of Alexandrov Surfaces with Curvature Bounded Below
In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.
Lipschitz regularity and approximate differentiability of the Diperna-Lions flow
Partial regularity of free discontinuity sets, II
A measure theoretic approach to higher codimension mean curvature flows
Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below - the Compact Case
Gradient flows with metric and differentiable structures, and applications to the Wasserstein space
In this paper we summarize some of the main results of a forthcoming book on this topic, where we examine in detail the theory of curves of maximal slope in a general metric setting, following some ideas introduced in [11, 5], and study in detail the case of the Wasserstein space of probability measures. In the first part we derive new general conditions ensuring convergence of the implicit time discretization scheme to a curve of maximal slope, the uniqueness, and the error estimates. In the second...
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