Dirichlet problem with $L^p$-boundary data in contractible domains of Carnot groups

Bonfiglioli, Andrea; Lanconelli, Ermanno

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 5 (2006) no. 4, pp. 579-610.

Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$. In this paper we study the Dirichlet problem for $ℒ$ with $L^p$-boundary data, on domains $\Omega$ which are contractible with respect to the natural dilations of $G$. One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$. A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Classification : 35J70, 35H20, 31B05, 31C15, 43A80

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