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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