On density of compactly supported smooth functions in fractional Sobolev spaces

We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $$W^{s,p}(\Omega )$$ W s , p ( Ω ) for an open, bounded set $$\Omega \subset \mathbb {R}^{d}$$ Ω ⊂ R d . The density property is closely related to the lower and upper Assouad codimension of the boundary of $$\Omega$$ Ω . We also describe explicitly the closure of $$C_{c}^{\infty }(\Omega )$$ C c ∞ ( Ω ) in $$W^{s,p}(\Omega )$$ W s , p ( Ω ) under some mild assumptions about the geometry of $$\Omega$$ Ω . Finally, we prove a variant of a fractional order Hardy inequality.

The extension problem for fractional Sobolev spaces with a partial vanishing trace condition

We construct whole-space extensions of functions in a fractional Sobolev space of order $$s\in (0,1)$$ s ∈ ( 0 , 1 ) and integrability $$p\in (0,\infty )$$ p ∈ ( 0 , ∞ ) on an open set O which vanish in a suitable sense on a portion D of the boundary $${{\,\mathrm{\partial \!}\,}}O$$ ∂ O of O. The set O is supposed to satisfy the so-called interior thickness condition in$${{\,\mathrm{\partial \!}\,}}O {\setminus } D$$ ∂ O \ D , which is much weaker than the global interior thickness condition. The proof works by means of a reduction to the case $$D=\emptyset $$ D = ∅ using a geometric construction.

Numerical approximations for fractional elliptic equations via the method of semigroups

We provide a novel approach to the numerical solution of the family of nonlocal elliptic equations (−Δ)su = f in Ω, subject to some homogeneous boundary conditions B on ∂Ω, where s ∈ (0,1), Ω ⊂ ℝn is a bounded domain, and (-Δ)s is the spectral fractional Laplacian associated to B on ∂Ω. We use the solution representation (−Δ)−s f together with its singular integral expression given by the method of semigroups. By combining finite element discretizations for the heat semigroup with monotone quadratures for the singular integral we obtain accurate numerical solutions. Roughly speaking, given a datum f in a suitable fractional Sobolev space of order r ≥ 0 and the discretization parameter h > 0, our numerical scheme converges as O(hr+2s), providing super quadratic convergence rates up to O(h4) for sufficiently regular data, or simply O(h2s) for merely f ∈ L2 (Ω). We also extend the proposed framework to the case of nonhomogeneous boundary conditions and support our results with some illustrative numerical tests.

THE DISTRIBUTIONAL -HESSIAN IN FRACTIONAL SOBOLEV SPACES

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

Functions of bounded higher variation in the fractional Sobolev spaces

In this paper, we establish fine properties of functions of bounded higher variation in the framework of fractional Sobolev spaces. In particular, inspired by the recent work of Brezis–Nguyen on the distributional Jacobian, we extend the definition of functions of bounded higher variation, which defined by Jerrard–Soner in [Formula: see text], to the fractional Sobolev space [Formula: see text], and apply Cartesian currents theory to establishing general versions of coarea formula, chain rule and decomposition property.

A critical fractional Choquard–Kirchhoff problem with magnetic field

In this paper, we are interested in a fractional Choquard–Kirchhoff-type problem involving an external magnetic potential and a critical nonlinearity [Formula: see text] [Formula: see text] where [Formula: see text] with [Formula: see text], [Formula: see text] is the Kirchhoff function, [Formula: see text] is the magnetic potential, [Formula: see text] is the fractional magnetic operator, [Formula: see text] is a continuous function, [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] and [Formula: see text] is the critical exponent of fractional Sobolev space. We first establish a fractional version of the concentration-compactness principle with magnetic field. Then, together with the mountain pass theorem, we obtain the existence of nontrivial radial solutions for the above problem in non-degenerate and degenerate cases.