Vekua theory for the Helmholtz operator

Moiola, A., Hiptmair, R. and Perugia, I. (2011) Vekua theory for the Helmholtz operator. Zeitschrift für Angewandte Mathematik und Physik, 62 (5). pp. 779-807. ISSN 0044-2275 doi: 10.1007/s00033-011-0142-3

Abstract/Summary

Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/28022
Identification Number/DOI	10.1007/s00033-011-0142-3
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Springer
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