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

Abstract

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.