Abstract
Conditions are presented under which properly elliptic second-order boundary value problems are well posed on irregular plane domains. The coefficients can be discontinuous. The results include known results for coercive forms, and also reduce to known results on proper ellipticity when the coefficients and domain are smooth. The main tool is an “inverse five-lemma” which relates the Neumann problem on a plane domain to a related modified Dirichlet problem. This inverse five-lemma can be used in a variety of settings. We show how it can be used to translate results of Grisvard on the index of Dirichlet operators in Sobolev spaces H s(Ω) to results on Neumann operators, and examine the implications for regularity.
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