Local density of Caputo-stationary functions in the space of smooth functions

We consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any $C^k\big([0,1]\big)$ function can be approximated in $[0,1]$ by a a function that is Caputo-stationary in $[0,1]$, with initial point $a<0$. Otherwise said, Caputo-stationary functions are dense in $C^k_{loc}(\mathbb{R})$.Comment: 19 pages, 4 figure

Alcune note sulle funzioni di seminorma frazionaria Ws,1 minima

In this survey we discuss some existence and asymptotic results, originally obtained in [4,3], for functions of least Ws,1-fractional seminorm. We present the connection between these functions and nonlocal minimal surfaces, leveraging this relation to build a function of least fractional seminorm. We further prove that a function of least fractional seminorm is the limit for p → 1 of the sequence of minimizers of the Ws,p-energy. Additionally, we consider the fractional 1-Laplace operator and study the equivalence between weak solutions and functions of least fractional seminorm.In questa nota discutiamo alcuni risultati di esistenza e asintotici, originariamente ottenuti in [4,3], per le funzioni di seminorma frazionaria Ws,1 minima. Presentiamo la connessione tra queste funzioni e le superfici minime nonlocali, e ricorriamo a tale relazione per costruire una funzione di seminorma frazionaria minima. Otteniamo inoltre una funzione di seminorma frazionario minima come limite per p → 1 del minimo dell'energia frazionaria Ws,p. Consideriamo in più l'1-Laplaciano frazionario e mostriamo l'equivalenza tra le soluzioni deboli e le funzioni di seminorma frazionaria Ws,1 minima

Some nonlocal operators and effects due to nonlocality

In this PhD thesis, we deal with problems related to nonlocal operators, in particular to the fractional Laplacian and to some other types of fractional derivatives (the Caputo and the Marchaud derivatives). We make an extensive introduction to the fractional Laplacian, we present some related contemporary research results and we add some original material. Indeed, we study the potential theory of this operator, introduce a new proof of Schauder estimates using the potential theory approach, we study a fractional elliptic problem in $\mathbb{R}^n$ with convex nonlinearities and critical growth and we present a stickiness property of nonlocal minimal surfaces for small values of the fractional parameter. Also, we point out that the (nonlocal) character of the fractional Laplacian gives rise to some surprising nonlocal effects. We prove that other fractional operators have a similar behavior: in particular, Caputo-stationary functions are dense in the space of smooth functions, moreover, we introduce an extension operator for Marchaud-stationary functions.Comment: 255 pages, 35 figures. PhD thesis Univ Milan (2017

Some observations on the Green function for the ball in the fractional Laplace framework

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new, however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.Comment: 45 pages, 2 figure

Rheological Effects of Some Xylanase on Doughs from High and Low Extraction Flours

AbstractThe xylanases are widely used in breadmaking with positive effects on bread quality but how they act in doughs is not fully understand yet. The aim of this study is to determine how different xylanolytic preparations modify the rheology of dough prepared from low and high extraction flours and the correlation between the rheological changes induced in dough and the viscosity and xylan content of flour extracts. Four flours, two white and two black, and three xylanolytic preparation was used in study. The rheological characteristics of dough were measured with the Extensograph. The xylan content and viscosity of flour extracts with xylanase were determined. In doughs from white flours xylanases increased the energy, maximum resistance and extensibility while in doughs from black flours decreased the energy and maximum resistance and increased the extensibility. The extensographic effects of xylanases were compared with their capacity to modify the viscosity and xylan content of aqueous flours extracts. The changes of extensographic indicators are well correlated with the changes of xylan content of extracts for white flour while for black flour correlations were observed with the changes of extracts viscosity. The capacity of xylanases to modify the viscosity of extract and convert the insoluble xylans in soluble xylans could be used to predict the performance of xylanases