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
Externally publishedYes

Fingerprint

Cone
Partial differential equation
Infinity
Dual Cone
Lower bound
Antipode
Closed
Irreducible polynomial
Partial Differential Operators
Convex Cone
Coefficient
Connected Components
Closure
Imply
Theorem

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone. / Murata, Minoru; Shibata, Yoshihiro.

In: Israel Journal of Mathematics, Vol. 31, No. 2, 06.1978, p. 193-203.

Research output: Contribution to journalArticle

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