Related papers: On a new class of fractional partial differential …

This paper presents a self-contained new theory of weak fractional differential calculus and fractional Sobolev spaces in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a…

This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one-dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural…

This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on…

For fractional derivatives and time-fractional differential equations, we construct a framework on the basis of the operator theory in fractional Sobolev spaces. Our framework provides a feasible extension of the classical Caputo and the…

In this paper, we give some properties and remarks of the new fractional Sobolev spaces with variable exponents. We also study the eigenvalue problem involving the new fractional $p(\cdot)$-Laplacian.

This paper presents a self-contained new theory of weak fractional differential calculus in one-dimension. The crux of this new theory is the introduction of a weak fractional derivative notion which is a natural generalization of integer…

Using the concept of fractional derivatives of Riemann$-$Liouville on time scales, we first introduce right fractional Sobolev spaces and characterize them. Then, we prove the equivalence of some norms in the introduced spaces, and obtain…

We consider a version of the fractional Sobolev inequality in domains and study whether the best constant in this inequality is attained. For the half-space and a large class of bounded domains we show that a minimizer exists, which is in…

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…

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type…

In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the…

In this paper, we introduce and study a new class of fractional modular function spaces, called \emph{Fractional Anisotropic Musielak--Sobolev Spaces}, which generalize both the fractional Anisotropic Orlicz--Sobolev spaces and the…

Based on recent developments in the theory of fractional Sobolev spaces, an interesting new class of nonlocal variational problems has emerged in the literature. These problems, which are the focus of this work, involve integral functionals…

We extend classical results on variational inequalities with convex sets with gradient constraint to a new class of fractional partial differential equations in a bounded domain with constraint on the distributional Riesz fractional…

Graph-based analysis holds both theoretical and applied significance, attracting considerable attention from researchers and yielding abundant results in recent years. However, research on fractional problems remains limited, with most of…

In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying first the nonlocal SPDEs and showing then the convergence of the family of these…

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 establish new Euclidean Sobolev logarithmic inequalities in the framework of fractional Sobolev spaces and their weighted version. Our approach relies on a interpolation inequality, which can be viewed as a fractional…

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least…