Related papers: Local Density of Solutions to Fractional Equations
The concept of local fractional derivative was introduced in order to be able to study the local scaling behavior of functions. However it has turned out to be much more useful. It was found that simple equations involving these operators…
In this paper, we present an overview of the development of one of the most dynamic areas of mathematics today: local differential operators of non-integer order. The underlying question is whether we are witnessing a period of…
The non-local problem is considered for the partial differential equation of mixed-type with Bessel operator and fractional order. An explicit solution is represented by Fourier-Bessel series in the given domain. It is established the…
Local solvability is analyzed for natural families of partial differential operators having double characteristics. In some families the set of all operators that are not locally solvable is shown to have both infinite dimension and…
Fractional difference operators possess nonlocal structure which largely affects and complicates the qualitative analysis of fractional difference equations. In this article, we discuss the effect of this memory property on asymptotic…
We prove some existence results for a class of nonlinear fractional equations driven by a nonlocal operator.
We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and…
In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the…
We investigate existence and uniqueness of solutions for a class of nonlinear nonlocal problems involving the fractional $p$-Laplacian operator and singular nonlinearities.
Fractional diffusion has become a fundamental tool for the modeling of multiscale and heterogeneous phenomena. However, due to its nonlocal nature, its accurate numerical approximation is delicate. We survey our research program on the…
We define, in a consistent way, non-local pseudo-differential operators acting on a space of analytic functionals. These operators include the fractional derivative case. In this context we show how to solve homogeneous and inhomogeneous…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small…
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…
The notion of a local fractional derivative (LFD) was introduced recently for functions of a single variable. LFD was shown to be useful in studying fractional differentiability properties of fractal and multifractal functions. It was…
It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and…
We discuss some basic properties of the eigenfunctions of a class of nonlocal operators whose model is the fractional p-Laplacian.
The purpose of this paper is three-fold: first, we survey on several known pointwise identities involving fractional operators; second, we propose a unified way to deal with those identities; third, we prove some new pointwise identities in…
This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and…