Basic properties of nonsmooth Hormander's vector fields and Poincare's inequality

We consider a family of vector fields defined in some bounded domain of R^p, and we assume that they satisfy Hormander's rank condition of some step r, and that their coefficients have r-1 continuous derivatives. We extend to this nonsmooth context some results which are well-known for smooth Hormander's vector fields, namely: some basic properties of the distance induced by the vector fields, the doubling condition, Chow's connectivity theorem, and, under the stronger assumption that the coefficients belong to C^{r-1,1}, Poincare's inequality. By known results, these facts also imply a Sobolev embedding. All these tools allow to draw some consequences about second order differential operators modeled on these nonsmooth Hormander's vector fields.Comment: 60 pages, LaTeX; Section 6 added and Section 7 (6 in the previous version) changed. Some references adde

Fractional Sobolev-Poincaré inequalities in irregular domains

This paper is devoted to the study of fractional (q, p)-Sobolev-Poincaré in- equalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincaré inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q, p)-Sobolev-Poincaré inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincaré implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out

The fate of the homoctenids (Tentaculitoidea) during the Frasnian-Famennian mass extinction (Late Devonian)

The homoctenids (Tentaculitoidea) are small, conical-shelled marine animals which are amongst the most abundant and widespread of all Late Devonian fossils. They were a principal casualty of the Frasnian-Famennian (F-F, Late Devonian) mass extinction, and thus provide an insight into the extinction dynamics. Despite their abundance during the Late Devonian, they have been largely neglected by extinction studies. A number of Frasnian-Famennian boundary sections have been studied, in Poland, Germany, France, and the United States. These sections have yielded homoctenids, which allow precise recognition of the timing of the mass extinction. It is clear that the homoctenids almost disappear from the fossil record during the latest Frasnian “Upper Kellwasser Event”. The coincident extinction of this pelagic group, and the widespread development of intense marine anoxia within the water column, provides a causal link between anoxia and the F-F extinction. Most notable is the sudden demise of a group, which had been present in rock-forming densities, during this anoxic event. One new species, belonging to Homoctenus is described, but is not formally named here

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Local invertibility in Sobolev spaces with applications to nematic elastomers and magnetoelasticity

We define a class of deformations in W^1,p(\u3a9,R^n), p>n 121, with positive Jacobian that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality between the distributional determinant and the pointwise determinant of the gradient. Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in W^1,p, and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity

Tentaculites from the Givetian and Frasnian of the Holy Cross Mountains

The Givetian and Frasnian strata of the Holy Cross Mountains, including chiefly biostromal sequence of the Kowala Formation, and the Dziewki Limestone of the Silesian Upland, yielded nine benthic and seven planktic tentaculite species. The tentaculites reveal strong affinities with coeval faunas of the East European Platform, and only two species from the oldest Givetian units (Stringocephalus-bearing strata), namely Homoctenus hanusi and Nowakia postotomari, are known primarily from the European Variscan belt. Stylioline succession allows recognition of the Middle/Late Devonian boundary in pelagic facies of the Kostomłoty area.Fauna tentakulitów z żywetu i franu Gór Świętokrzyskich i antykliny Siewierza wykazują duże analogie z jednowiekową fauną Europy Wschodniej (głównie Platformy Rosyjskiej), a jedynie stratygraficznie najstarsze gatunki (przede wszystkim Homoctenus hanusi), dowodzą związków z dewonem Czech i Niemiec. Wykazano znaczenie tych mięczaków dla korelacji utworów żywetu i franu oraz zarysowano możliwość wykorzystania sekwencji styliolin do rozpoznawania granicy żywetu z franem w basenowych facjach strefy kostomłockiej