On the monotonicity of perimeter of convex bodies

Let $n\ge2$ and let $\Phi\colon\mathbb{R}^n\to[0,\infty)$ be a positively $1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in $\mathbb{R}^n$, the monotonicity of anisotropic $\Phi$-perimeters holds, i.e. $P_\Phi(A)\le P_\Phi(B)$. In this note, we prove a quantitative lower bound on the difference of the $\Phi$-perimeters of $A$ and $B$ in terms of their Hausdorff distance.Comment: 8 page

Improved Lipschitz approximation of HH-perimeter minimizing boundaries

We prove two new approximation results of $H$-perimeter minimizing boundaries by means of intrinsic Lipschitz functions in the setting of the Heisenberg group $\mathbb{H}^n$ with $n\ge2$. The first one is an improvement of a result of Monti and is the natural reformulation in $\mathbb{H}^n$ of the classical Lipschitz approximation in $\mathbb{R}^n$. The second one is an adaptation of the approximation via maximal function developed by De Lellis and Spadaro.Comment: 25 page

A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up

We introduce the new space $BV^{\alpha}(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha \in (0, 1)$ via a new distributional approach exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical $BV$ theory, we give a new notion of set $E$ of (locally) finite fractional Caccioppoli $\alpha$-perimeter and we define its fractional reduced boundary $\mathscr{F}^{\alpha} E$. We are able to show that $W^{\alpha,1}(\mathbb{R}^n)\subset BV^\alpha(\mathbb{R}^n)$ continuously and, similarly, that sets with (locally) finite standard fractional $\alpha$-perimeter have (locally) finite fractional Caccioppoli $\alpha$-perimeter, so that our theory provides a natural extension of the known fractional framework. Our main result partially extends De Giorgi's Blow-up Theorem to sets of locally finite fractional Caccioppoli $\alpha$-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets.Comment: 46 page

Sobolev subspaces of nowhere bounded functions

We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elements are nowhere bounded functions. We also prove the existence of a closed infinite dimensional linear subspace whose non zero elements are nowhere Lq functions for suitable values of q larger than the Sobolev exponent

Problèmes et Perspectives des finances locales en Italie

Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 dofinansowane zostało ze środków MNiSW w ramach działalności upowszechniającej naukę

Generalized Bakry-\'Emery curvature condition and equivalent entropic inequalities in groups

We study a generalization of the Bakry-\'Emery pointwise gradient estimate for the heat semigroup and its equivalence with some entropic inequalities along the heat flow and Wasserstein geodesics for metric-measure spaces with a suitable group structure. Our main result applies to Carnot groups of any step and to the $\mathbb{SU}(2)$ group.Comment: 76 page

A distributional approach to fractional Sobolev spaces and fractional variation

In this thesis, we present the distributional approach to fractional Sobolev spaces and fractional variation developed in [20, 22, 23]. The new space BV\u1d45(\u211d\u207f) of functions with bounded fractional variation in \u211d\u207f of order \u3b1 08 (0, 1) is distributionally defined by exploiting suitable notions of fractional gradient and fractional divergence already existing in the literature. In analogy with the classical BV theory, we give a new notion of set E of (locally) finite fractional Caccioppoli \u3b1-perimeter and we define its fractional reduced boundary F\u1d45E. We are able to show that W\u1d45,\ub9(\u211d\u207f) 82 BV \u1d45(\u211d\u207f) continuously and, similarly, that sets with (locally) finite standard fractional \u3b1-perimeter have (locally) finite fractional Caccioppoli \u3b1-perimeter, so that our theory provides a natural extension of the known fractional framework. We first extend De Giorgi\u2019s Blow-up Theorem to sets of locally finite fractional Caccioppoli \u3b1-perimeter, proving existence of blow-ups and giving a first characterisation of these (possibly non-unique) limit sets. We then prove that the fractional \u3b1-variation converges to the standard De Giorgi\u2019s variation both pointwise and in the \u393-limit sense as \u3b1 \u2192 1- and, similarly, that the fractional \u3b2-variation converges to the fractional \u3b1-variation both pointwise and in the \u393-limit sense as \u3b2 \u2192 \u3b1- for any given \u3b1 08 (0, 1). Finally, by exploiting some new interpolation inequalities on the fractional operators involved, we prove that the fractional \u3b1-gradient converges to the Riesz transform as \u3b1 \u2192 0\u207a in Lp for p 08 (1,+ 1e) and in the Hardy space and that the \u3b1-rescaled fractional \u3b1-variation converges to the integral mean of the function as \u3b1 \u2192 0\u207a

On the monotonicity of weighted perimeters of convex bodies

We prove that, among weighted isotropic perimeters, only constant multiples of the Euclidean perimeter satisfy the monotonicity property on nested convex bodies. Although the analogous result fails for general weighted anisotropic perimeters, a similar characterization holds for radially-weighted anisotropic densities.Comment: 7 page

Gli Annessi ICAO: in particolare sul dovere degli Stati di notificare le discordanze tra i regolamenti interni e gli standards (art. 38 Convenzione di Chicago)

L'emanazione dei regolamenti tecnico-giuridici, denominati Annessi, il cui scopo è quello di uniformare norme e procedure tra gli Stati contraenti è una delle più importanti manifestazioni dell'Organizzazione Internazionale dell'Aviazione Civile. La Convenzione di Chicago è stata resa esecutiva in Italia con decreto legislativo 6 marzo 1948 n. 616, ratificato con la legge 17 aprile 1956 n.561, l'attuazione dell'impegno a recepire gli Annessi (che è conseguente alla ratifica della Convenzione) è stato più lungo ed articolato, infatti nel tempo si sono susseguiti numerosi interventi legislativi. Il lavoro analizza l'attuazione degli standards le pratiche raccomandate, la responsabilità degli Stati, le differenze dall’Annesso 14. In conclusione viene esaminata la posizione degli USA relativamente alla regolamentazione ICAO e la programmazione di questi ultimi per l’incremento del traffico aereo

Non-local BVBV functions and a denoising model with L1L^1 fidelity

We study a general total variation denoising model with weighted $L^1$ fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel $K$, and the approximation term is given by the $L^1$ norm with respect to a non-singular measure with positively lower-bounded $L^\infty$ density. We provide a detailed analysis of the space of non-local $BV$ functions with finite total $K$-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the $K$-variation and the associated $K$-perimeter. Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the study of the fidelity in our model.Comment: 32 pages, see v2 for longer versio