Some remarks on the Sobolev inequality in Riemannian manifolds

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe's type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the $p$-hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.Comment: 16 page

Optimal decay rate for degenerate parabolic equations on noncompact manifolds

We consider an initial value problem for a doubly degenerate parabolic equation in a noncompact Riemannian manifold. The geometrical features of the manifold are coded in either a Faber-Krahn inequality or a relative Faber-Krahn inequality. We prove optimal decay and space-time local estimates of solutions. We employ a simplified version of the by now classical local approach by DeGiorgi, Ladyzhenskaya-Uraltseva, DiBenedetto which is of independent interest even in the euclidean case

Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modelled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and non-negative. We show that the asymptotic profile for large times of u is one-dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem) we obtain a genuine multi-dimensional profile given by the well known Barenblatt solution

Evolution and architecture of a West Mediterranean Upper Pleistocene to Holocene coastal apron-fan system

The Quaternary deposits of tectonically stable areas are a powerful tool to investigate high-frequency climate variations (<10 ka) and to distinguish allogenic and autogenic factors controlling deposition. Therefore, an Upper Pleistocene—Holocene coastal apron-fan system in north—western Sardinia (Porto Palmas, Italy) was studied to investigate the relations between climate changes, sea-level fluctuations and sediment source-supply that controlled its development. The sedimentary sequence records the strong influence of local (wet/dry) and worldwide (sea-level) environmental variations in the sedimentation and preservation of the deposits. A multi-disciplinary approach allowed subdivision of the succession into four major, unconformity-bounded stratigraphic units: U1 U2, U3 and U4. Unit U1, tentatively dated to the warm and humid Marine Isotopic Stage (MIS) 5, consists of sandy, gravelly coastal/beach deposits developed during high sea-level in low-lying areas. Unit U2 consists of debris-flow dominated fan-deposits (ca 74 ka; MIS 4), preserved as partial fills of small valleys and coves. Unit U2 is mainly composed of reddish silty conglomerate to pebbly siltstones sourced from the Palaeozoic metamorphic inland hills (bedrock), superficially disintegrated during the preceding warm, vegetation-rich MIS 5. The cold and semi-arid climate strongly reduced vegetation cover along the valley flanks. Therefore, sediment gravity-flow processes, possibly activated by rainstorms, led to deposition of debris-flow dominated fans. Unit U3 consists of water-flow dominated alluvial-fan deposits (ca 47 to 23 ka; MIS 3), developed on a slightly inclined coastal plain. Unit U3 is composed of sandstone and sandy conglomerate fed from two main sediment sources: metamorphic inland bedrock and Quaternary bioclastic-rich shelf-derived sands. During this cold phase, sea-level dropped sufficiently to expose bioclastic sands accumulated on the shelf. Frequent climate fluctuations favoured inland aeolian transport of sand during dry phases, followed by reworking of the aeolian bodies by flash floods during wet phases. Bedrock-derived fragments mixed with water-reworked, wind-blown sands led to the development of water-flow dominated fans. The Dansgaard—Oeschger events possibly associated with sand landward deflation and main fan formations are Dansgaard—Oeschger 13 (ca 47 ka), Dansgaard—Oeschger 8 (ca 39 ka) and Dansgaard—Oeschger 2 (ca 23 ka). No record of sedimentation during MIS 2 was observed. Finally, bioclastic-rich aeolianites (Unit U4, ca 10 to 5 ka; MIS 1), preserved on a coastal slope, were developed during the Holocene transgression (ca 10 to 5 ka; MIS 1). The studied sequence shows strong similarities with those of other Mediterranean sites; it is, however, one of the few where the main MIS 4 and MIS 3 climatic fluctuations are registered in the sedimentary record

Modeling environmental responses of plantassociations by fuzzy set theory

A method for studying the response of vegetation to environmental gradients, based on the community niche and fuzzy set theory, is presented. The approach is illustrated using an example from perennial halophilous vegetation along the Northern Adriatic coast of Italy. Compatibility curves are obtained by fuzzy set theoretical methods, and are used tomodel the response functions of plant associations to environmental gradients, including soil and ground water salinity, soil pH, soil and ground water temperature, percentage of sand, and variations in the ground water level. The compatibility curves summarize the similarity of a given plant community, with a particular value of an environmental variable, to the species combination of a given plant association. Compatibility curves offer an alternative approach to non-linear regression and best fit analyses normally used to model single species responses to environmental gradients. The approach is particularly useful given there is no singlemechanisticmodel that can capture the exact shape of the functional response along environmental gradients, and given that environmental data are commonly affected by high levels of noise