Soliton dynamics for a general class of Schr\"odinger equations

The soliton dynamics for a general class of nonlinear focusing Schr\"odinger problems in presence of non-constant external (local and nonlocal) potentials is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation.Comment: 26 page

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

An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework

An Eigenvalue Problem for Nonlocal Equations

In this paper we study the existence of a positive weak solution for a class of nonlocal equations under Dirichlet boundary conditions and involving the regional fractional Laplacian operator...Our result extends to the fractional setting some theorems obtained recently for ordinary and classical elliptic equations, as well as some characterization properties proved for differential problems involving different elliptic operators. With respect to these cases studied in literature, the nonlocal one considered here presents some additional difficulties, so that a careful analysis of the fractional spaces involved is necessary, as well as some nonlocal L^q estimates, recently proved in the nonlocal framework

Asymptotically linear problems driven by fractional Laplacian operators

In this paper we study a non-local fractional Laplace equation, depending on a parameter, with asymptotically linear right-hand side. Our main result concerns the existence of weak solutions for this equation and it is obtained using variational and topological methods. We treat both the nonresonant case and the resonant one

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