### 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 language | English |
---|---|

Pages (from-to) | 193-203 |

Number of pages | 11 |

Journal | Israel Journal of Mathematics |

Volume | 31 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1978 Jun |

Externally published | Yes |

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### 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.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 31, no. 2, pp. 193-203. https://doi.org/10.1007/BF02760551

}

TY - JOUR

T1 - Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

AU - Murata, Minoru

AU - Shibata, Yoshihiro

PY - 1978/6

Y1 - 1978/6

N2 - 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.

AB - 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.

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U2 - 10.1007/BF02760551

DO - 10.1007/BF02760551

M3 - Article

AN - SCOPUS:51249184907

VL - 31

SP - 193

EP - 203

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 2

ER -