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On the uniqueness theorem of Holmgren
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-4971-7147
2015 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 281, no 1-2, 357-378 p.Article in journal (Refereed) Published
Abstract [en]

We review the classical Cauchy-Kovalevskaya theorem and the related uniqueness theorem of Holmgren, in the simple setting of powers of the Laplacian and a smooth curve segment in the plane. As a local problem, the Cauchy-Kovalevskaya and Holmgren theorems supply a complete answer to the existence and uniqueness issues. Here, we consider a global uniqueness problem of Holmgren's type. Perhaps surprisingly, we obtain a connection with the theory of quadrature identities, which demonstrates that rather subtle algebraic properties of the curve come into play. For instance, if is the interior domain of an ellipse, and I is a proper arc of the ellipse , then there exists a nontrivial biharmonic function u in which is three-flat on I (i.e., all partial derivatives of u of order vanish on I) if and only if the ellipse is a circle. Another instance of the same phenomenon is that if is bounded and simply connected with -smooth Jordan curve boundary, and if the arc is nowhere real-analytic, then we have local uniqueness already with sub-Cauchy data: if a function is biharmonic in for some planar neighborhood of I, and is three-flat on I, then it vanishes identically on , provided that is connected. Finally, we consider a three-dimensional setting, and analyze it partially using analogues of the square of the standard Cauchy-Riemann operator. In a special case when the domain is of periodized cylindrical type, we find a connection with the massive Laplacian [the Helmholz operator with imaginary wave number] and the theory of generalized analytic (or pseudoanalytic) functions of Bers and Vekua.

Place, publisher, year, edition, pages
2015. Vol. 281, no 1-2, 357-378 p.
Keyword [en]
Cauchy problem, Dirichlet problem, Holmgren's uniqueness theorem
National Category
URN: urn:nbn:se:kth:diva-173414DOI: 10.1007/s00209-015-1488-6ISI: 000359830400015ScopusID: 2-s2.0-84939575710OAI: diva2:853822

QC 20150915

Available from: 2015-09-15 Created: 2015-09-11 Last updated: 2015-09-15Bibliographically approved

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Hedenmalm, Håkan
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