Weak and viscosity solutions of the fractional Laplace equation

Aim of this paper is to give a regularity result for weak solutions of a fractional Laplacian equation. In order to get this result we first prove a maximum principle and then, using it, an interior and boundary regularity result for weak solutions of the problem. As a consequence of our regularity result, along the paper we also deduce that the first eigenfunction of the fractional Laplacian is strictly positive

H^s versus C^0-weighted minimizers

We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C^0-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H^s-topology.Comment: 15 page

Dandy-Walker malformation: is the "tail sign" the key sign?

OBJECTIVE.To demonstrate the value of the "tail sign" in the assessment of Dandy-Walker Malformation (DWM). METHODS: A total of 31fetal MRI, performed before 24 weeks of gestation after second-line US examination between May 2013 and September 2014, were examined retrospectively. All MRI examinations were performed using a 1.5 Tesla magnet without maternal sedation. RESULTS: MRI diagnosed 15/31 cases of Dandy-Walker Malformation, 6/31 cases of vermian partial caudal agenesis, 2/31 of vermian hypoplasia, 4/31 of vermian malrotation, 2/31 of Walker-Warburg Syndrome, 1/31 of Blake pouch cyst, 1/31 of rhombencephalosynapsis. All data were compared with fetopsy results, Fetal MR after the 30th week or postnatal MRI; the follow up depended on the maternal decision to terminate or continue pregnancy. In our review study we found the presence of the "tail sign"; this sign was visible only in Dandy-Walker Malformation and Walker-Warburg Syndrome. CONCLUSION: The "tail sign" could be helpful in the difficult differential diagnosis between Dandy Walker, vermian malrotation, vermian hypoplasia and vermian partial agenesis

Nonlinear problems on the Sierpi\'nski gasket

This paper concerns with a class of elliptic equations on fractal domains depending on a real parameter. Our approach is based on variational methods. More precisely, the existence of at least two non-trivial weak (strong) solutions for the treated problem is obtained exploiting a local minimum theorem for differentiable functionals defined on reflexive Banach spaces. A special case of the main result improves a classical application of the Mountain Pass Theorem in the fractal setting, given by Falconer and Hu (1999)

All functions are (locally) ss-harmonic (up to a small error) - and applications

The classical and the fractional Laplacians exhibit a number of similarities, but also some rather striking, and sometimes surprising, structural differences. A quite important example of these differences is that any function (regardless of its shape) can be locally approximated by functions with locally vanishing fractional Laplacian, as it was recently proved by Serena Dipierro, Ovidiu Savin and myself. This informal note is an exposition of this result and of some of its consequences

Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains

We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples

Toward a more expressive pattern matching in Haskell

LAUREA MAGISTRALEIl Pattern matching è diventato molto importante nella programmazione funzionale. Molti linguaggi in questa categoria ne dipendono pesantemente, soprattutto per quanto riguarda la definizione di funzioni. In questo caso il pattern matching assume il ruolo di dispatcher ed esegue la giusta porzione di codice in base alla struttura dei parametri. Programmare con questo paradigma pattern-action, porta ad ottenere programmi meno soggetti a errori. Ma quanto potenti possono essere i pattern in modo che sempre più logica sia espressa al loro interno piuttosto che nel codice che verrà eseguito? Molti linguaggi cercano di rendere i pattern più espressivi e potenti. Ne è un esempio F# che rende i pattern estensibili con gli active patterns. In questa tesi analizzo il pattern matching di Haskell, un linguaggio puramente funzionale, cercando i suoi punti di forza e debolezza. Una volta identificati, descrivo un'estensione che sfrutta la potenza del suo pattern matching, e ne cerca di superare alcuni limiti. L'estensione è definita come un superset della sintassi di Haskell, in questo modo ogni programma scritto nella forma estesa può essere compilato in Haskell.Pattern matching has become a very important feature in functional programming. Many languages in this category heavily depend on it, especially when it comes to the definition of functions. In this case, pattern matching acts like a dispatcher that executes the right portion of code depending on the structure of the arguments. Coding with this pattern-action paradigm leads to less error-prone programs. But how powerful can patterns be, so that always more logic is expressed with them rather than in the code that will be executed? Many languages try to make patterns more expressive and powerful. One example is F# that makes the pattern system extensible with its active patterns. In this thesis, I analyze pattern matching in Haskell, a purely functional programming language, finding its strengths and weaknesses. Once identified those, I describe an extension that takes advantage of the power of its pattern matching and tries to overcome some of its limits. This extension is defined as a superset of the Haskell syntax, this means that every program written in the extended form can be compiled down to Haskell