The fractional Cheeger problem

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.Comment: 33 pages, 2 figure

Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems

We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset \real^N$ is a bounded domain, and $1 \leq q < 2$. For $q=1$, $|u|^{q-2}u$ will be identified with \sgn(u). We establish a variational principle for least energy nodal solutions, and we investigate their qualitative properties. In particular, we show that they satisfy a unique continuation property (their zero set is Lebesgue-negligible). Moreover, if $\Omega$ is radial, then least energy nodal solutions are foliated Schwarz symmetric, and they are nonradial in case $\Omega$ is a ball. The case $q=1$ requires special treatment since the formally associated energy functional is not differentiable, and many arguments have to be adjusted

The second eigenvalue of the fractional p−p-Laplacian

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\lambda_2(\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem $$\inf \{\lambda_2(\Omega)\,:\,|\Omega|=c\}.$$ We prove that, differently from the local case, an optimal shape does not exist, even among disconnected sets. A minimizing sequence is given by the union of two disjoint balls of volume $c/2$ whose mutual distance tends to infinity.Comment: 38 pages. The test function used in the proof of Theorem 3.1 needed to be slightly modified, in order to be admissible for $1<p<2$. We fixed this issu

On the existence threshold for positive solutions of p-laplacian equations with a concave-convex nonlinearity

We study the following boundary value problem with a concave-convex nonlinearity: \begin{equation*} \left\{ \begin{array}{r c l l} -\Delta_p u & = & \Lambda\,u^{q-1}+ u^{r-1} & \textrm{in }\Omega, \\ u & = & 0 & \textrm{on }\partial\Omega. \end{array}\right. \end{equation*} Here $\Omega \subset \mathbb{R}^n$ is a bounded domain and $1<q<p<r<p^*$. It is well known that there exists a number $\Lambda_{q,r}>0$ such that the problem admits at least two positive solutions for $0<\Lambda<\Lambda_{q,r}$, at least one positive solution for $\Lambda=\Lambda_{q,r}$, and no positive solution for $\Lambda > \Lambda_{q,r}$. We show that $$\lim_{q \to p} \Lambda_{q,r} = \lambda_1(p),$$ where $\lambda_1(p)$ is the first eigenvalue of the p-laplacian. It is worth noticing that $\lambda_1(p)$ is the threshold for existence/nonexistence of positive solutions to the above problem in the limit case $q=p$

"Omne, quod no est ex fide, peccatum est". The relevance of good faith in canonical transactio

Il substrato canonistico della transazione è di particolare evidenza. In essa, infatti, il conflitto ha spesso già raggiunto lo stadio litigioso. Ragion per cui nella sua interpretazione la dottrina tende a separare i due profili economico ed etico. Su quest’ultimo, in particolare, la Chiesa, pur dotata di un sistema di sanzioni giuridiche necessariamente imperfetto, ha svolto, nei secoli, un ruolo di primo piano. É infatti un insegnamento evangelico che alla base di ogni rapporto umano debba regnare la concordia, poiché concordia mater est unitatis: se le parti, ignorando l’etica, stanno per giungere ad una lite, o vi siano giunte, la Chiesa deve esortarle a comporla. Ciò che si intende qui indagare è dunque la rilevanza di due fra i requisiti essenziali dell’istituto transattivo, ovvero la lis e la res dubia, all’interno del sistema delle Decretali (X 1.36. 1-11 de transactionibus). La cornice sarà offerta dal principio cardine che sancisce l’effettività del negozio: Effectus transactionis est, ut ei stetur; questo per valutare come debba essere intesa nell’ordinamento canonico l’incertezza della lite in rapporto al principio di buona fede.It is quite clear that there is a canonical foundation underlying the institute of transactio. Indeed, a compromise is often reached when a dispute has already entered the litigation phase, and as such legal doctrine tends to separate the economic aspect from the ethical aspect in its interpretation. Though the Church has a necessarily imperfect system of legal sanctions at its disposal, over the centuries it has in fact played a fundamental role in the ethical aspect of compromise. Indeed, the Gospel teaches that every human relationship must be based on concordia, as concordia mater est unitatis: if parties have ignored such morals and are about to litigate, or have already started legal proceedings, the Church must exhort them to settle the dispute. This article aims to examine the relevance of two of the essential requirements of transactio – namely lis and res dubia – within the Decretals (X 1.36. 1-11 de transactionibus). The framework of this analysis is provided by the tenet Effectus transactionis est, ut ei stetur, which sanctions the effectiveness of transactio as a juristic act. The goal is to evaluate how the uncertainty of legal proceedings relates to the principle of good faith in canon law.Ciencias ReligiosasDerech

A Designer A Day 2013

A designer A day Il famoso detto “una mela al giorno …” si trasforma in “un designer/progetto al giorno”, da cui il titolo dell’iniziativa. Una mostra/evento di una settimana, durante il Solone del Mobile con il supporto de La Trentina. Un progetto di selezione abbinato ad un concorso internazionale realizzato in modalità last minutes. Giovani designer, selezionati da un’apposita giuria, hanno avuto l’opportunità di essere protagonisti per un giorno presentando un proprio progetto all’interno di uno spazio espositivo preallestito e personalizzabile. I temi di questa edizione erano "autoproduzione" e "vivere sostenibile". Sono stati selezionati 11 progetti di designer/autoproduttori italiani e stranieri: Sanserif Creatius & Sara Sorribes – Moments Collettivo Buscai-_-minimal-Crist Pace, Forcellini, Mennella – Pronto intervento Filippo Franchi – Sedia Walter DesignSottoscala_Affumicatore_02 Ivdesign – Appendino Francesca Romei – GR_EGG _egg separator P. UNO – ITER MINO_copertina_Giovanni_Tomasini_DESIGN Giovanni Tomasini – Mino Laura Rovida – Eco intrecci Simone Guida – Evoluzione di forme vegetal