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HOLOMORPHIC EXTENSION OF SOLUTIONS OF ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS AND A COMPLEX HUYGENS' PRINCIPLE

Published online by Cambridge University Press:  01 February 1997

PETER EBENFELT
PETER EBENFELT
Affiliation:
Department of Mathematics, Royal Institute of Technology, 100 44 Stockholm, Sweden. E-mail: [email protected]
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Abstract

We show that solutions of analytic elliptic partial differential equations of the form

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in a simply connected domain Ω ⊂ ℝn can be extended holomorphically into [Nscr ] (Ω) ⊂ [Copf ]n, the so-called kernel of the harmonicity hull of Ω. This extends results of Avanissian, Aronszajn etc., on polyharmonic functions and also results of Vekua, Khavinson and Shapiro in ℝ2. We also find the domain of influence for solutions of a certain subclass of these operators, in terms of their Cauchy data on analytic hypersurfaces in [Copf ]n (a complex Huygens' principle). As an application, we investigate reflection properties of these solutions and, in particular, solutions of the Helmholtz equation in ℝ3.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 87 - 104
Copyright
The London Mathematical Society 1997

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