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.

A fractional version of Rivière’s GL(n)-gauge

We prove that for antisymmetric vector field $$\Omega $$ Ω with small $$L^2$$ L 2 -norm there exists a gauge $$A \in L^\infty \cap {\dot{W}}^{1/2,2}({\mathbb {R}}^1,GL(N))$$ A ∈ L ∞ ∩ W ˙ 1 / 2 , 2 ( R 1 , G L ( N ) ) such that $$\begin{aligned} {\text {div}}_{\frac{1}{2}} (A\Omega - d_{\frac{1}{2}} A) = 0. \end{aligned}$$ div 1 2 ( A Ω - d 1 2 A ) = 0 . This extends a celebrated theorem by Rivière to the nonlocal case and provides conservation laws for a class of nonlocal equations with antisymmetric potentials, as well as stability under weak convergence.

Deformation and Hochschild cohomology of coisotropic algebras

Coisotropic algebras consist of triples of algebras for which a reduction can be defined and unify in a very algebraic fashion coisotropic reduction in several settings. In this paper, we study the theory of (formal) deformation of coisotropic algebras showing that deformations are governed by suitable coisotropic DGLAs. We define a deformation functor and prove that it commutes with reduction. Finally, we study the obstructions to existence and uniqueness of coisotropic algebras and present some geometric examples.

Existence of an optimal domain for the buckling load of a clamped plate with prescribed volume

We formulate the minimization of the buckling load of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small, we show that the optimal shape problem with prescribed volume is solved. In addition, we discuss two different choices for the penalization term.