The Bernstein Problem for Embedded Surfaces in the Heisenberg Group H

In the paper [13] we proved that the only stable C 2 minimal surfaces in the first Heisenberg group H 1 which are graphs over some plane and have empty characteristic locus must be vertical planes. This result represents a sub-Riemannian version of the celebrated theorem of Bernstein. In this paper we extend the result in [13] to C 2 complete em-bedded minimal surfaces in H 1 with empty characteristic locus. We prove that every such a surface without boundary must be a vertical plane. This result represents a sub-Riemannian coun-terpart of the classical theorems of Fischer-Colbrie and Schoen, [16], and do Carmo and Peng, [15], and answers a question posed by Lei Ni

The extension problem for Sobolev spaces on the Heisenberg group

We prove that if a domain $\Omega$ on the Heisenberg group \IH\sp{n} satisfies the $(\epsilon ,\delta)$ condition then there is a linear bounded extension operator ${\cal E}$ from {\cal L}\sp{k,p}(\Omega ) into {\cal L}\sp{k,p}(\IH\sp{n}) where $1\le k,\ 1\le p\le\infty$

On the best possible character of the LQ norm in some a priori estimates for non-divergence form equations in Carnot groups

Let G \boldsymbol {G} be a group of Heisenberg type with homogeneous dimension Q Q . For every 0 > ϵ > Q 0>\epsilon >Q we construct a non-divergence form operator L ϵ L^\epsilon and a non-trivial solution u ϵ ∈ L 2 , Q − ϵ ( Ω ) ∩ C ( Ω ¯ ) u^\epsilon \in \mathcal {L}^{2,Q-\epsilon }(\Omega )\cap C(\overline {\Omega }) to the Dirichlet problem: L u = 0 Lu=0 in Ω \Omega , u = 0 u=0 on ∂ Ω \partial \Omega . This non-uniqueness result shows the impossibility of controlling the maximum of u u with an L p L^p norm of L u Lu when p > Q p>Q . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as $$sup Ω u ≤ C ( ∫ Ω | det ⁡ ( u , i j ) | d g ) 1 / m , \sup _\Omega u\le C\left (\int _{\Omega }|\operatorname {det}(u_{,ij})|\,dg\right ) ^{1/m},$$ where m m is the dimension of the horizontal layer of the Lie algebra and ( u , i j ) (u_{,ij}) is the symmetrized horizontal Hessian of u u

MUTUAL ABSOLUTE CONTINUITY OF HARMONIC AND SURFACE MEASURES FOR HÖRMANDER TYPE OPERATORS

In this paper we study the Dirichlet problem for the sub-Laplacian associated with a system X = {X1,...,Xm} of C ∞ real vector fields in R n satisfying Hörmander’s finite rank condition (1.1) rank Lie[X1,...,Xm] ≡ n. Throughout this paper n ≥ 3, and X ∗ j denotes the formal adjoint of Xj. The sub-Laplacia