Israel Journal of Mathematics

, Volume 16, Issue 1, pp 103–116 | Cite as

Lower bounds at infinity for solutions of differential equations with constant coefficients

  • Lars Hörmander


Two extensions of a classical theorem of Rellich are proved: (1) LetP=P(−iϖ/ϖx) be a partial differential operator with constant coefficients in\(\mathbb{R}^n \), let the manifolds contained in\(\left\{ {\xi \in \mathbb{R}^n ;P(\xi ) = 0} \right\}\) have codimension ≧k>0, and denote by Γ an open cone in\(\mathbb{R}^n \) intersecting each normal plane of every such manifold. If
,Pu=0 and
it follows thatu=0. (2) Assume in addition that each irreducibe lfactor ofP van shes on a real hypersurface and that Γ contains both normal directions at some such point. If
andP(D) u has compact support, the same condition withk=1 implies thatu has compact support. In both results the hypotheses on the cone Γ and on the operatorP are minimal.


Compact Support Real Hypersurface Real Zero Partial Differential Operator Open Cone 
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Copyright information

© Hebrew University 1973

Authors and Affiliations

  • Lars Hörmander
    • 1
  1. 1.Mathematiska InstitutionenLunds UniversitetLundSweden

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