We report on recent progress in the study of nonlinear diffusion equations involving
nonlocal, long-range diffusion effects. Our main concern is the so-called fractional porous …

We study the regularity up to the boundary of solutions to the Dirichlet problem for the
fractional Laplacian. We prove that if u is a solution of (− Δ) su= g in Ω, u≡ 0 in R n\Ω, for …

A recently developed nonlocal vector calculus is exploited to provide a variational analysis
for a general class of nonlocal diffusion problems described by a linear integral equation on …

We show global uniqueness in an inverse problem for the fractional Schrödinger equation:
an unknown potential in a bounded domain is uniquely determined by exterior …

We describe the mathematical theory of diffusion and heat transport with a view to including
some of the main directions of recent research. The linear heat equation is the basic …

Progress in Mathematics is a series of books intended for professional mathematicians and
scientists, encompassing all areas of pure mathematics. This distinguished series, which …

In this paper, we develop a novel finite difference method to discretize the fractional
Laplacian (− Δ) α/2 in hypersingular integral form. By introducing a splitting parameter, we …

Fractional derivatives can be used to model time delays in a diffusion process. When the
order of the fractional derivative is distributed over the unit interval, it is useful for modeling a …

Lévy flights and Lévy walks serve as two paradigms of random walks resembling common
features but also bearing fundamental differences. One of the main dissimilarities is the …

We study solutions to L u= f in Ω⊂ R n, being L the generator of any, possibly non-
symmetric, stable Lévy process. On the one hand, we study the regularity of solutions to L u …