Asymptotic analysis for radial sign-changing solutions of the Brezis-Nirenberg problem

We study the asymptotic behavior, as $\lambda \rightarrow 0$, of least energy radial sign-changing solutions $u_\lambda$, of the Brezis-Nirenberg problem \begin{equation*} \begin{cases} -\Delta u = \lambda u + |u|^{2^* -2}u & \hbox{in}\ B_1\\ u=0 & \hbox{on}\ \partial B_1, \end{cases} \end{equation*} where $\lambda >0$, $2^*=\frac{2n}{n-2}$ and $B_1$ is the unit ball of $\R^n$, $n\geq 7$. We prove that both the positive and negative part $u_\lambda^+$ and $u_\lambda^-$ concentrate at the same point (which is the center) of the ball with different concentration speeds. Moreover we show that suitable rescalings of $u_\lambda^+$ and $u_\lambda^-$ converge to the unique positive regular solution of the critical exponent problem in $\R^n$. Precise estimates of the blow-up rate of $\|u_\lambda^\pm\|_{\infty}$ are given, as well as asymptotic relations between $\|u_\lambda^\pm\|_{\infty}$ and the nodal radius $r_\lambda$. Finally we prove that, up to constant, $\lambda^{-\frac{n-2}{2n-8}} u_\lambda$ converges in $C_{loc}^1(B_1-\{0\})$ to $G(x,0)$, where $G(x,y)$ is the Green function of the Laplacian in the unit ball.Comment: 28 page

Sign-changing bubble-tower solutions to fractional semilinear elliptic problems

We study the asymptotic and qualitative properties of least energy radial sign-changing solutions to fractional semilinear elliptic problems of the form $$\begin{cases} (-\Delta)^s u = |u|^{2^*_s-2-\varepsilon}u &\text{in } B_R, \\ u = 0 &\text{in }\mathbb{R}^n \setminus B_R, \end{cases}$$ where $s \in (0,1)$, $(-\Delta)^s$ is the s-Laplacian, $B_R$ is a ball of $\mathbb{R}^n$, $2^*_s := \frac{2n}{n-2s}$ is the critical Sobolev exponent and $\varepsilon>0$ is a small parameter. We prove that such solutions have the limit profile of a "tower of bubbles", as $\varepsilon \to 0^+$, i.e. the positive and negative parts concentrate at the same point with different concentration speeds. Moreover, we provide information about the nodal set of these solutions

A nonexistence result for sign-changing solutions of the Brezis-Nirenberg problem in low dimensions

We consider the Brezis-Nirenberg problem: \begin{equation*} \begin{cases} -\Delta u = \lambda u + |u|^{2^* -2}u & \hbox{in}\ \Omega\\ u=0 & \hbox{on}\ \partial \Omega, \end{cases} \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$, $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda>0$ a positive parameter. The main result of the paper shows that if $N=4,5,6$ and $\lambda$ is close to zero there are no sign-changing solutions of the form $$u_\lambda=PU_{\delta_1,\xi}-PU_{\delta_2,\xi}+w_\lambda,$$ where $PU_{\delta_i}$ is the projection on $H_0^1(\Omega)$ of the regular positive solution of the critical problem in $\mathbb{R}^N$, centered at a point $\xi \in \Omega$ and $w_\lambda$ is a remainder term. Some additional results on norm estimates of $w_\lambda$ and about the concentrations speeds of tower of bubbles in higher dimensions are also presented.Comment: 21 page

Spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space

In this paper we investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space $\mathbb{L}^{n+1}$, for $n\geq 2$, spanning a given boundary datum lying on the hyperbolic space $\mathbb{H}^n$

Interprete di comandi vocali come ausilio per persone diversamente abili

Questo lavoro di Tesi è finalizzato allo sviluppo di un dispositivo portatile a comando vocale per il controllo tramite infrarossi di apparati domestici; il dispositivo è espressamente rivolto a persone diversamente abili, in particolare ai disabili motori agli arti superiori. E’ stato perseguito l’obiettivo di progettare un dispositivo che possa presentare basso prezzo rispetto agli analoghi ausili esistenti, disponendo contestualmente di funzioni di ausilio presenti solo in dispositivi di fascia alta. E' stata scelta la tecnologia di riconoscimento vocale Speaker Dependent, e individuato un opportuno modulo hardware commerciale che la implementa. Nella maggior parte dei dispositivi stand-alone di riconoscimento vocale, risulta necessario affiancare a tale hardware un dispositivo host per la gestione di altre parti e funzioni del dispositivo. E’ stato sviluppato un protocollo di comunicazione per l’interazione fra un generico host e il modulo di riconoscimento vocale. Il firmware sul modulo di riconoscimento vocale è stato sviluppato in modo da consentirne l’utilizzo come periferica di riconoscimento Speaker Dependent per impieghi generali. E’ stata sviluppata e implementata su PC l’architettura firmware/software dell’host per l’applicazione oggetto di questo lavoro. Tale architettura gestisce gli aspetti generali dell’interazione con la periferica di riconoscimento vocale e implementa le funzioni di ausilio all’utente diversabile richieste da specifica. Sono state soddisfatte le specifiche sul numero e l'organizzazione dei comandi vocali gestibili dal sistema. E’ stato totalmente progettato e parzialmente realizzato l’hardware del sistema telecomando. E' stata soddisfatta la specifica sul costo del sistema

A sharp gradient estimate and W2,qW^{2,q} regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq 3$. We first prove a new gradient estimate for classical solutions with smooth data $\rho$. As a consequence we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class $W^{2,q}$ and strictly spacelike in $\mathbb{R}^N$, provided that $\rho\in L^q(\mathbb{R}^N) \cap L^m(\mathbb{R}^N)$ with $q>N$ and $m\in[1,\frac{2N}{N+2}]$

Linking forest diversity and tree health: preliminary insights from a large-scale survey in Italy

Forest health is currently assessed in Europe (ICP Forests monitoring program). Crown defoliation and dieback, tree mortality, and pathogenic damage are the main aspects considered in tree health assessment. The worsening of environmental conditions (i.e., increase of temperature and drought events) may cause large-spatial scale tree mortality and forest decline. However, the role of stand features, including tree species assemblage and diversity as factors that modify environmental impacts, is poorly considered. The present contribution reanalyses the historical dataset of crown conditions in Italian forests from 1997 to 2014 to identify ecological and structural factors that influence tree crown defoliation, highlighting in a special manner the role of tree diversity. The effects of tree diversity were explored using the entire data set through multivariate cluster analyses and on individual trees, analysing the influence of the neighbouring tree diversity and identity at the local (neighbour) level. Preliminary results suggest that each tree species shows a specific behaviour in relation to crown defoliation, and the distribution of crown defoliation across Italian forests reflects the distribution of the main forest types and their ecological equilibrium with the environment. The potentiality and the problems connected to the possible extension of this analysis at a more general level (European and North American) were discussed

Salivary pH, calcium, phosphorus and selected enzymes in healthy dogs: A pilot study

Abstract Background Saliva in dogs, as in humans, is a complex fluid secreted by different salivary glands in the oral cavity to protect the oral mucosa and teeth. The use of saliva as a substitute for blood in diagnosing and prognosticating disease in humans is widely accepted. Salivary biochemistry has also been used as a marker for periodontal disease in humans. No studies have as yet investigated the relation between salivary biochemistry and periodontal disease in dogs, however; neither has the salivary composition of healthy dogs with no oral disease been assessed. The purpose of this study was to obtain an overview on pH distribution and a set of salivary biochemical analytes (calcium, phosphorus, lactate dehydrogenase, lysozyme and amylase) commonly related to oral health in humans in a subset population of healthy young dogs with no periodontal disease or previous oral disease. Data were analyzed to gather salivary reference ranges for pH and each parameter and to assess a possible correlation between salivary and serum analytes. Results Twenty-nine adult client-owned dogs were recruited for the study. Lactate dehydrogenase and lysozyme showed higher concentrations in saliva than in serum, whereas amylase showed the contrary. Salivary biochemistry values did not differ between males and females or between non-neutered and neutered individuals. No significant correlations between salivary and serum calcium, phosphorus, lactate dehydrogenase, amylase and lysozyme were identified in this study. Data allowed intervals for the salivary pH and other analytes investigated to be obtained from healthy dogs with healthy oral conditions. Conclusions These preliminary data can contribute to enlarge our understanding of the functional role of saliva and its relation to oral health in dogs