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The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity. (English) Zbl 0725.31004
Differential equations and their applications, Proc. 7th Conf., Equadiff 7, Prague/Czech. 1989, Teubner-Texte Math. 118, 171-174 (1990).
[For the entire collection see Zbl 0704.00019.]
Given a sublaplacian L on a stratified Lie algebra $${\mathcal N}$$ of type I, geometric criteria for the regularity (irregularity resp.) of open subsets are given. Assuming that $${\mathcal N}$$ is of step r and that $${\mathcal N}$$ is identified with the underlying space $${\mathbb{R}}^ m$$, such an operator is of the form $$L=X^ 2_ 1+...+X^ 2_ n$$ with smooth vectorfields $$X_ i$$ which together with their commutators up to order r span the tangent space at each point of $${\mathbb{R}}^ m$$. In terms of Euclidean geometry the following results are obtained: If $$n=2$$ and $$m>2$$ then an exterior (r/2-$$\epsilon$$)-Hölder condition is not sufficient for U to be regular. Whenever $$n\geq 3$$ and $$m>3$$, even an exterior (r- $$\epsilon$$)-condition is not sufficient. The only cases where all bounded domains with smooth boundary are regular are the cases $$r\leq 2$$ and the few cases where $$n=2$$ and $$r\leq 4$$. The results generalize the well known cone conditions for the classical Laplacian.
Reviewer: W.Hansen
##### MSC:
 31D05 Axiomatic potential theory 35J25 Boundary value problems for second-order elliptic equations