On a new class of fractional partial differential equations

Abstract

Abstract. In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend to many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.

A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up

“…For an account on the existing literature on the operator ∇ α , see [31,Section 1]. Here we only refer to [29][30][31][32][33][35][36][37] for the articles tightly connected to the present work and to [27,Section 15.2] for an agile presentation of the fractional operators defined in (1.2) and in (1.4) and of some of their elementary properties. According to [33,Section 1], it is interesting to notice that [20] seems to be the earliest reference for the operator defined in (1.4).…”

“…In the forthcoming paper [9], it will be proved that also the inclusion S α,p (R n ) ⊂ L α,p (R n ) holds continuously, so that the spaces S α,p (R n ) and L α,p (R n ) coincide. In particular, we get the following relations: S α+ε,p (R n ) ⊂ W α,p (R n ) ⊂ S α−ε,p (R n ) with continuous embeddings for all α ∈ (0, 1), p ∈ (1, +∞) and 0 < ε < min{α, 1 − α}, see [32,Theorem 2.2]; S α,2 (R n ) = W α,2 (R n ) for all α ∈ (0, 1), see [32,Theorem 2.2]; W α,p (R n ) ⊂ S α,p (R n ) with continuous embedding for all α ∈ (0, 1) and p ∈ (1,2], see [38,Chapter V,Section 5.3].…”

Analysis and Application of a Nonlocal Hessian

“…The precise form we have chosen for the non-local Hessian can be derived from the model case of nonlocal gradients -the fractional gradient -which has been developed in [SS14]. Here we prove several results analogous to the first order case, as in [MS14], for the generalizations involving generic radial weights that satisfy (1.2)-(1.4).…”

Some remarks on rearrangement for nonlocal functionals

“…Recently, a notion of fractional gradients (more precisely distributional Riesz fractional gradients) has been introduced in the literature by Shieh and Spector in the papers [31,32], where several basic properties of local Sobolev spaces (e.g. Sobolev, Morrey, Hardy, Trudinger inequalities) are proven to extend to fractional spaces defined through this new notion.…”

“…for a suitable normalization constant C N,s depending on N and s. Notice also that [31,32] According to [31,32], one can define, for p > 1 and s ∈ (0, 1), the space …”