This paper is concerned with qualitative properties of solutions to nonlocal reaction–diffusion equations of the form 0true∫RN∖KJ(x−y)0.16em(ufalse(yfalse)−ufalse(xfalse))0.16emnormaldy+f(ufalse(xfalse))=0,1emx∈RN∖K, set in a perforated open set double-struckRN∖K, where K⊂double-struckRN is a bounded compact ‘obstacle’ and f is a bistable nonlinearity. When K is convex, we prove some Liouville‐type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on K.

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the formwhere K ⊂ R N is a bounded compact set, called an "obstacle", and f is a bistable nonlinearity. When K is convex, it is known that solutions ranging in [0, 1] and satisfying u(x) → 1 as |x| → ∞ must be identically 1 in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data f and J for which this property fails.

We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces W 1,p derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the BV (R N ), W 1,p (R N ), B s p,∞ (R N ) and C 0,1 (R N ) scales and we obtain new equivalent characterizations for these spaces. We also establish a noncompactness result for sequences and new (non-)limiting embeddings between Lipschitz and Besov spaces which extend the previous known results.2010 Mathematics Subject Classification. 46E35.

In this paper, we study the smoothness of restrictions of Besov functions. It is known that for any f ∈ B s p,q (R N ) with q p we have f (·, y) ∈ B s p,q (R d ) for a.e. y ∈ R N −d . We prove that this is no longer true when p < q. Namely, we construct a functionWe show that, in fact, f (·, y) belong to B (s,Ψ) p,q (R d ) for a.e. y ∈ R N −d , a Besov space of generalized smoothness, and, when q = ∞, we find the optimal condition on the function Ψ for this to hold. The natural generalization of these results to Besov spaces of generalized smoothness is also investigated. Contents 29References 29 2010 Mathematics Subject Classification. 35J50.