Alcuni risultati sulla regolarità dei vettori analitici-Gevrey

My aim is to give, in this talk, some topics on the question of regularity of Analytic-Gevrey vectors of partial differential operators (p.d.o.) with analytic-Gevrey coefficients. Since the results obtained in the sixties on elliptic p.d.o's, which are both hypoelliptic (C∞ setting), analytic-Gevrey hypoelliptic (analytic-Gevrey setting) and satisfy the so-called Kotake-Narasimhan property, a lot of works and articles were devoted to these problems in case of non elliptic p.d.o's under suitable hypotheses (for example on the degeneracy of ellipticity). I will consider the third problem on analytic-Gevrey vectors in the three cases of global (on compact manifolds), local (near a point in the base-space), microlocal (near a point in the cotangent space), situations, and say few words on the main two methods used in order to obtain positive (or negative) results. Finally I will focus on some new microlocal results on degenerate elliptic (also called sub-elliptic) p.d.o's of second order, obtained in a common work with Gregorio Chinni.Il mio scopo è quello di trattare, in questo intervento, alcuni argomenti sulla regolarità dei vettori analitici-Gevrey di operatori differenziali alle derivate parziali (p.d.o.) a coefficienti analitici-Gevrey.Nella parte finale mi concentrerò su alcuni nuovi risultati microlocali relativi a p.d.o. degeneri ellittici (anche detti sub-ellittici) del secondo ordine, ottenuti in un lavoro con Gregorio Chinni

Diverse salinity responses in Crithmum maritimum tissues at different salinities over time

Crithmum maritimum (sea fennel) withstands high salinity, and to better understand how different protective mechanisms against salinity are activated, young seedlings were exposed to increasing concentrations of NaCl (0 to 512 mM) over six weeks. Plant survival and chlorophyll content were reduced at >85 mM NaCl and growth was affected at > 341 mM NaCl. Relative water content fell and Na+ accumulated more in leaves than in roots. Induction of Na+/H+ antiporter expression reached a maximum at 427 mM NaCl in both tissues. Salinity induced the accumulation of proline, soluble sugars and glycine betaine. All three accumulated to higher levels in leaves than roots and greatest accumulation was after 6 weeks and the highest salt concentrations. Hydrogen peroxide levels fell with increasing salinity in leaves, while ascorbic acid and catalase activity rose. Overall, the most dramatic changes occurred after six weeks of saline stress but different mechanisms were activated at different salinity thresholds and in the two tissues. Key salinity thresholds in the response of Crithmum maritimum to salinity stress are identified activating different mechanisms. At 85 mM NaCl roots reach osmotic adjustment, at 171 mM further osmolyte protection mechanisms are activated, at 256 mM NaCl leaves reach osmotic adjustment, at 341 mM plant growth is affected and at the highest salinity tested, 512 mM, protective mechanisms are affected in leaves but not in roots

Multiple mechanisms mediate growth and survival in young seedlings of two populations of the halophyte Atriplex halimus (L) subjected to long single step salinity treatments

Understanding how halophytes survive high soil salinity in realistic long-term experiments is important for strategies to mitigate the effects of increasing soil salinity worldwide. Protective mechanisms in halophytes enabling survival include sequestration of salt via Na+/H+ antiporters, synthesis and accumulation of osmolytes, and activation of protective mechanisms against reactive oxygen species (ROS). Protective mechanisms elicited by a single step-up to a range of NaCl treatments (34–256 mM) in two populations of the halophyte Atriplex halimus L. from contrasting environments (arid steppe and saline coastline) were compared over 6 weeks. The coastal population survived significantly better at high salinity compared with the steppe population, although in both populations, salinity inhibited growth. Increased Na+ and K+ concentration was accompanied by higher induction of Na+/H+ antiporter gene expression in coastal than in steppe population leaves. Osmolytes increased more significantly in the coastal than in the steppe population with greater induction of choline mono-oxygenase gene expression. Activation of ROS scavenging mechanisms was greater in coastal than in steppe plants. Differential responses found through time, in different salt concentrations, and between leaves and roots indicate a finely tuned response. Sharp changes in responses at 171 mM NaCl indicate that different mechanisms may be invoked at different stress levels.</jats:p

The Dirichlet problem for superdegenerate differential operators

Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of H\"ormander, we show that the Dirichlet problem associated with $L$ has a unique smooth classical solution. The proof uses the Malliavin calculus. At present, there appears to be no proof of this result using classical analytic techniques