Another way to say subsolution: The maximum principle and sums of Green functions

R. S. Laugesen, N. A. Watson

Research output: Contribution to journalArticle

Abstract

Consider an elliptic second order differential operator L with no zeroth order term (for example the Laplacian L = -Δ). If Lu ≤ 0 in a domain U, then of course u satisfies the maximum principle on every subdomain V ⊂ U. We prove a converse, namely that Lu ≤ 0 on U if on every subdomain V, the maximum principle is satisfied by u + v whenever v is a finite linear combination (with positive coefficients) of Green functions with poles outside V. This extends a result of Crandall and Zhang for the Laplacian. We also treat the heat equation, improving Crandall and Wang's recent result. The general parabolic case remains open.

Original languageEnglish (US)
Pages (from-to)127-153
Number of pages27
JournalMathematica Scandinavica
Volume97
Issue number1
DOIs
StatePublished - Jan 1 2005

ASJC Scopus subject areas

  • Mathematics(all)

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