On isoperimetric inequalities with respect to infinite measures

We study isoperimetric problems with respect to infinite measures on $R ^n$. In the case of the measure $\mu$ defined by $d\mu = e^{c|x|^2} dx$, $c\geq 0$, we prove that, among all sets with given $\mu-$measure, the ball centered at the origin has the smallest (weighted) $\mu-$perimeter. Our results are then applied to obtain Polya-Szego-type inequalities, Sobolev embeddings theorems and a comparison result for elliptic boundary value problems.Comment: 25 page

Quantum tricriticality in transverse Ising-like systems

The quantum tricriticality of d-dimensional transverse Ising-like systems is studied by means of a perturbative renormalization group approach focusing on static susceptibility. This allows us to obtain the phase diagram for 3<d<4, with a clear location of the critical lines ending in the conventional quantum critical points and in the quantum tricritical one, and of the tricritical line for temperature T \geq 0. We determine also the critical and the tricritical shift exponents close to the corresponding ground state instabilities. Remarkably, we find a tricritical shift exponent identical to that found in the conventional quantum criticality and, by approaching the quantum tricritical point increasing the non-thermal control parameter r, a crossover of the quantum critical shift exponents from the conventional value \phi = 1/(d-1) to the new one \phi = 1/2(d-1). Besides, the projection in the (r,T)-plane of the phase boundary ending in the quantum tricritical point and crossovers in the quantum tricritical region appear quite similar to those found close to an usual quantum critical point. Another feature of experimental interest is that the amplitude of the Wilsonian classical critical region around this peculiar critical line is sensibly smaller than that expected in the quantum critical scenario. This suggests that the quantum tricriticality is essentially governed by mean-field critical exponents, renormalized by the shift exponent \phi = 1/2(d-1) in the quantum tricritical region.Comment: 9 pages, 2 figures; to be published on EPJ

Finsler Hardy inequalities

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved.Comment: 31 pages. We add "Note added to Proof" in Introduction and several reference

Singular elliptic equations having a gradient term with natural growth

We study a class of Dirichlet boundary value problems whose prototype is \begin{equation}\label{1.2abs} \left\{\begin{array}{ll} -\Delta_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ \Omega\,,\\ u\ge 0\,,&{\quad\hbox{in } \ \Omega}\\ u = 0\,&\quad\hbox{on }\partial \Omega\,,\end{array}\right. \end{equation} where $\Omega$ an open bounded subset of $\mathbb R^N$, $0<q<1$, $1<p<N$, $h$ is a continuous function and $f$ belongs to a suitable Lebesgue space. The main features of this problem are the presence of a singular term and a first order term with natural growth in the gradient. A priori estimates and existence results are proved depending on the summability of the datum $f$

Neural networks for driver behavior analysis

The proliferation of info-entertainment systems in nowadays vehicles has provided a really cheap and easy-to-deploy platform with the ability to gather information about the vehicle under analysis. With the purpose to provide an architecture to increase safety and security in automotive context, in this paper we propose a fully connected neural network architecture considering positionbased features aimed to detect in real-time: (i) the driver, (ii) the driving style and (iii) the path. The experimental analysis performed on real-world data shows that the proposed method obtains encouraging results

The isoperimetric problem for a class of non-radial weights and applications

We study a class of isoperimetric problems on R+ N where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type |x|kxN α. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities

On the behaviour of the solutions to pp-Laplacian equations as pp goes to 11

In the present paper we study the behaviour as $p$ goes to $1$ of the weak solutions to the problems $$\begin{cases} -\operatorname{div} \bigl(|\nabla u_p|^{p-2}\nabla u_p\bigr)=f &\text{in } \Omega\\ u_p=0 &\text{on } \partial\Omega, \end{cases}$$ where $\Omega$ is a bounded open set of ${\mathbb R}^N$ $(N\ge 2)$ with Lipschitz boundary and p > 1. As far as the datum $f$ is concerned, we analyze several cases: the most general one is $f\in W^{-1,\infty}(\Omega)$. We also illustrate our results by means of remarks and examples