A fractional Hadamard formula and applications

We derive a shape derivative formula for the family of principal Dirichlet eigenvalues $$\lambda _s(\Omega )$$ λ s ( Ω ) of the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s associated with bounded open sets $$\Omega \subset \mathbb {R}^N$$ Ω ⊂ R N of class $$C^{1,1}$$ C 1 , 1 . This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case $$s=\frac{1}{2}$$ s = 1 2 . As an application, we consider the maximization problem for $$\lambda _s(\Omega )$$ λ s ( Ω ) among annular-shaped domains of fixed volume of the type $$B\setminus \overline{B}'$$ B \ B ¯ ′ , where B is a fixed ball and $$B'$$ B ′ is ball whose position is varied within B. We prove that $$\lambda _s(B\setminus \overline{B}')$$ λ s ( B \ B ¯ ′ ) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings.

Compact Sobolev embeddings on non-compact manifolds via orbit expansions of isometry groups

Given a complete non-compact Riemannian manifold (M, g) with certain curvature restrictions, we introduce an expansion condition concerning a group of isometries G of (M, g) that characterizes the coerciveness of G in the sense of Skrzypczak and Tintarev (Arch Math 101(3): 259–268, 2013). Furthermore, under these conditions, compact Sobolev-type embeddings à la Berestycki-Lions are proved for the full range of admissible parameters (Sobolev, Moser-Trudinger and Morrey). We also consider the case of non-compact Randers-type Finsler manifolds with finite reversibility constant inheriting similar embedding properties as their Riemannian companions; sharpness of such constructions are shown by means of the Funk model. As an application, a quasilinear PDE on Randers spaces is studied by using the above compact embeddings and variational arguments.

Dynamics of stochastic retarded Benjamin-Bona-Mahony equations on unbounded channels

<p style='text-indent:20px;'>This article is devoted to the asymptotic behaviour of solutions for stochastic Benjamin-Bona-Mahony (BBM) equations with distributed delay defined on unbounded channels. We first prove the existence, uniqueness and forward compactness of pullback random attractors (PRAs). We then establish the forward asymptotic autonomy of this PRA. Finally, we study the non-delay stability of this PRA. Due to the loss of usual compact Sobolev embeddings on unbounded domains, the forward uniform tail-estimates and forward flattening of solutions are used to prove the forward asymptotic compactness of solutions.</p>