Some Variants of the Poincaré-Lemma

Abstract

AMS(MOS) 35J20, 26A86 The classical Poincare-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 Maulen [2] proved that for general (unbounded) domains in Rn, , the statement of this Lemma stays true if one replaces where the weight-function pδ grows as . In this paper I first extend his result to domains ω contained in Rn with and 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 (with general n), whose complements contain a hyperplane, it suffices to take and for general domains in it suffices to take . 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.