Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

Minoru Murata, Yoshihiro Shibata

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone in R n and -Γ′ be the antipodes of the dual cone of Γ. Let {Mathematical expression} be a partial differential operator with constant coefficients in R n, where Q(ζ)≠0 on R n-iΓ′ and P i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R n-iΓ′;P j(ζ)=0, grad P j(ζ)≠0} contains some real point on which grad P j≠0 and grad P j∉Γ∪(-Γ). Let C be an open cone in R n-Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R n;P(ξ)=0}. If u∈ℒ′∩L loc 2 (R n-Γ) and the support of P(-i∂/∂x)u is contained in Γ, then the condition {Mathematical expression} implies that the support of u is contained in Γ.

Original languageEnglish
Pages (from-to)193-203
Number of pages11
JournalIsrael Journal of Mathematics
Volume31
Issue number2
DOIs
Publication statusPublished - 1978 Jun 1
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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