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The Dirichlet problem for sublaplacians on nilpotent Lie groups - geometric criteria for regularity. (English) Zbl 0601.31007
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- \(\epsilon)\)-Hölder 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.

MSC:
31D05 Axiomatic potential theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
22E25 Nilpotent and solvable Lie groups
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