The Bernstein problem in Heisenberg groups

In these notes, we collect the main and, to the best of our knowledge, most up-to-date achievements concerning the Bernstein problem in the Heisenberg group; that is, the problem of determining whether the only entire minimal graphs are hyperplanes. We analyze separately the problem for t-graphs and for intrinsic graphs: in the first case, the Bernstein Conjecture turns out to be false in any dimension, and a complete characterization of minimal graphs is available in H1 for the smooth case. A positive result is instead available for Lipschitz intrinsic graphs in H1; moreover, one can see that the conjecture is false in Hn with n at least 5, by adapting the Euclidean counterexample in high dimension; the problem is still open when n is 2, 3 or 4

Γ\Gamma-convergence for functionals depending on vector fields. II. Convergence of minimizers

Given a family of locally Lipschitz vector fields $X(x)=(X_1(x),\dots,X_m(x))$ on $\mathbb{R}^n$, $m\leq n$, we study integral functionals depending on $X$. Using the results in \cite{MPSC1}, we study the convergence of minima, minimizers and momenta of those functionals. Moreover, we apply these results to the periodic homogenization in Carnot groups and to prove a $H$-compactness theorem for linear differential operators of the second order depending on $X$

Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric

We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given

Classical flows of vector fields with exponential or sub-exponential summability

We show that vector fields $b$ whose spatial derivative $D_xb$ satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if $D_xb$ satisfies a suitable exponential summability condition then the flow associated to $b$ has Sobolev regularity, without assuming boundedness of ${\rm div}_xb$. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations.Comment: 35 page

Discovery of a radio relic in the low mass, merging galaxy cluster PLCK G200.9-28.2

Radio relics at the peripheries of galaxy clusters are tracers of the elusive cluster merger shocks. We report the discovery of a single radio relic in the galaxy cluster PLCK G200.9-28.2 ($z=0.22$, $M_{500} = 2.7\pm0.2 \times 10^{14} M_{\odot}$) using the Giant Metrewave Radio Telescope at 235 and 610 MHz and the Karl G. Jansky Very Large Array at 1500 MHz. The relic has a size of $\sim 1 \times 0.28$ Mpc, an arc-like morphology and is located at 0.9 Mpc from the X-ray brightness peak in the cluster. The integrated spectral index of the relic is $1.21\pm0.15$. The spectral index map between 235 and 610 MHz shows steepening from the outer to the inner edge of the relic in line with the expectation from a cluster merger shock. Under the assumption of diffusive shock acceleration, the radio spectral index implies a Mach number of $3.3\pm1.8$ for the shock. The analysis of archival XMM Newton data shows that PLCK G200.9-28.2 consists of a northern brighter sub-cluster, and a southern sub-cluster in a state of merger. This cluster has the lowest mass among the clusters hosting single radio relics. The position of the Planck Sunyaev Ze'ldovich effect in this cluster is offset by 700 kpc from the X-ray peak in the direction of the radio relic, suggests a physical origin for the offset. Such large offsets in low mass clusters can be a useful tool to select disturbed clusters and to study the state of merger.Comment: 10 pages, 7 figures, 4 tables. Accepted for publication in MNRA

Infliximab in recalcitrant severe atopic eczema associated with contact allergy.

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Poincaré-type inequality for Lipschitz continuous vector fields

open4noG. C. and M. M. are partially supported by MAnET Marie Curie Initial Training Networks (ITN). A. P. was supported by the Progetto CaRiPaRo “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems” and now is supported by ERC ADG GeMeThNES n∘ 246923 and GNAMPA of INDAM. F. S.C. is supported by MIUR, Italy, GNAMPA of INDAM, University of Trento and MAnET Marie Curie Initial Training Networks (ITN) n∘ 607643.The scope of this paper is to prove a Poincaré type inequality for a family of nonlinear vector fields, whose coefficients are only Lipschitz continuous with respect to the distance induced by the vector fields themselves.openCitti, Giovanna; Manfredini, Maria; Pinamonti, Andrea; Serra Cassano, FrancescoCitti, Giovanna; Manfredini, Maria; Pinamonti, Andrea; Serra Cassano, Francesc