Second order stability for the Monge-Ampere equation and strong Sobolev convergence of Optimal Transport Maps

The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in W2,1loc if their right hand side converge strongly in L1loc. As a corollary we deduce strong W1,1loc stability of optimal transport maps

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures

BV Estimates in Optimal Transportation and Applications

In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well and we also provide a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an L∞ bound, where we prove that the total variation decreases by projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, we obtain BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also establish some properties of the Wasserstein projection which are interesting in their own right, and allow, for instance, for the proof of the uniqueness of such a projection in a very general framework

Capsular polysaccharides of cultured phototrophic biofilms

Phototrophic biofilm samples from an Italian wastewater treatment plant were studied in microcosm experiments under varying irradiances, temperatures and flow regimes to assess the effects of environmental variables and phototrophic biomass on capsular exopolysaccharides (CPS). The results, obtained from circular dichroism spectroscopy and High Performance Liquid Chromatography, suggest that CPS have a stable spatial conformation and a complex monosaccharide composition. The total amount present was positively correlated with the biomass of cyanobacteria and diatoms, and negatively with the biovolume of green algae. The proportion of uronic acids showed the same correlation with these taxon groups, indicating a potential role of cyanobacteria and diatoms in the removal of residual nutrients and noxious cations in wastewater treatment. While overall biofilm growth was limited by low irradiance, high temperature (30 degrees C) and low flow velocity (25 l h(-1)) yielded the highest phototrophic biomass, the largest amount of CPS produced, and the highest proportion of carboxylic acids present

Characterization of exopolysaccharides produced by seven biofilm-forming cyanobacterial strains for biotechnological applications

The molecular identification of seven biofilmforming cyanobacteria and the characterization of their exopolysaccharides were made and considered in terms of potential biotechnological applications. The studied strains were isolated from phototrophic biofilms taken from various Italian sites including a wastewater treatment plant, an eroded soil, and a brackish lagoon. The polysaccharides were characterized by use of ion exchange chromatography, circular dichroism, and cytochemical stains. All strains produced exopolysaccharides with differing ratios of hydrophobic and hydrophilic moieties depending on the species, the polysaccharide fraction (i.e., whether capsular or released), and the ambient conditions. It was shown that the anionic nature of the exopolysaccharides was due to the presence of carboxylic and sulfated groups and is likely the main characteristic with industrial applicability. Potential biotechnological applications are discusse

Cyanoflan: A cyanobacterial sulfated carbohydrate polymer with emulsifying properties

Abstract The extracellular polysaccharides produced by cyanobacteria have distinctive characteristics that make them promising for applications ranging from bioremediation to biomedicine. In this study, a sulfated polysaccharide produced by a marine cyanobacterial strain and named cyanoflan was characterized in terms of morphology, chemical composition, and rheological and emulsifying properties. Cyanoflan has a 71% carbohydrate content, with 11% of sulfated residues, while the protein account for 4% of dry weight. The glycosidic-substitution analysis revealed a highly branched complex chemical structure with a large number of sugar residues. The cyanoflan high molecular mass fractions (above 1 MDa) and entangled structure is consistent with its high apparent viscosity in aqueous solutions and high emulsifying activity. It showed to be a typical non-Newtonian fluid with pseudoplastic behavior. Altogether, these results confirm that cyanoflan is a versatile carbohydrate polymer that can be used in different biotechnological applications, such as emulsifying/thickening agent in food or cosmetic industries