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A representation theorem for solutions of the Helmholtz equation and resolvent estimates for the Laplacian. (English) Zbl 0704.35018

Analysis, et cetera, Res. Pap. in Honor of J. Moser’s 60th Birthd., 39-76 (1990).
[For the entire collection see Zbl 0688.00009.]
The author proves a representation theorem for solutions of the Helmholtz equation \(\Delta u+k^ 2u=0\) in \({\mathbb{R}}^ n\) where k is any complex number \(\neq 0\) (Theorem 4.3). This implies a complete characterization of those solutions which have a representation of the form \[ u(x)=\int_{S^{n-1}}\exp (ikx\cdot \omega)\phi (\omega)d\omega \] with some \(\phi \in L^ 2(S^{n-1})\). The result follows from certain resolvent estimates for functions of the form \(-(\Delta +k^ 2)^{-1}\) f for k in the upper halfplane (Section 3).
Reviewer: G.Džiuk

MSC:

35C15 Integral representations of solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P25 Scattering theory for PDEs

Citations:

Zbl 0688.00009