Abstract
AMS(MOS) 35J20, 26A86
The classical Poincaré-Lemma states, that for every bounded, open set ω contained in
exists a constant K such that for all functions ƒ with compact support in ω
Mäulen [2] proved that for general (unbounded) domains in R
n,
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, the statement of this Lemma stays true if one replaces
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where the weight-function pδ grows as
![](//:0)
. In this paper I first extend his result to domains ω contained in R
n with
![](//:0)
and
![](//:0)
by a new method of proof. Then I shall employ this method to get two new versions of the “generalized Poincare-Lemma” with slower growing weight-functions. For domains in
![](//:0)
(with general n), whose complements contain a hyperplane, it suffices to take
![](//:0)
and for general domains in
![](//:0)
it suffices to take
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.
As applications I treat boundary-value problems for the Laplacian on unbounded domains, where I get very simple and explicit a priori-estimates for the norm of the solution operator.
Additional information
Notes on contributors
Rainer Janssen
Wolfgang Walter