The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations

In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Caratheodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simpler class of CC spaces, i.e. the Heisenberg group H^n. In particular we will positively answer to the Bernstein problem in the case n=1 and we will provide counterexamples when n>=5

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

Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness, smoothness'

We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solutio

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

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