Good morning everybody.


I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ and an horizontal distribution spanned by $X=\partial\_x-\frac y2\partial\_z$ and $Y=\partial\_y+\frac x2 \partial\_z$.


I don't look for representation formulas, rather to the standard existence results like harnack maximum principles, etc..


Many thanks in advance.


Regards, Guido



dirichlet problem in the heisenberg group
I would look at:



> 
> A. Bonfiglioli, E. Lanconelli, F. Uguzzoni. *Stratified Lie Groups
>  and Potential Theory
>  for their Sub-Laplacians*. Springer, 2007. [DOI 10.1007/978-3-540-71897-0](http://dx.doi.org/10.1007/978-3-540-71897-0)
> 
> 
> 


This book is always my first reference for this sort of general theory. In Section 7.1 you can find a theorem giving sufficient conditions on an open set $U$ for the Dirichlet problem $Lu=0$ on $U$, $u = \varphi$ on $\partial U$, to be solvable (here I presume you want $L = X^2 + Y^2$ to be the usual sub-Laplacian); e.g. Proposition 7.1.5. The book has a lot more material on the potential theory of $L$ as well. And in Chapter 5 you can find a development of Harnack inequalities, maximum principles, mean value theorems, etc, for functions (sub-)harmonic with respect to $L$.


