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 Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is the interior domain of an ellipse, and I is a proper arc of the ellipse ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document}, then there exists a nontrivial biharmonic function u in Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} which is three-flat on I (i.e., all partial derivatives of u of order ≤2\documentclass[12pt]{minimal}
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\begin{document}$$\le 2$$\end{document} vanish on I) if and only if the ellipse is a circle. Another instance of the same phenomenon is that if Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is bounded and simply connected with C∞\documentclass[12pt]{minimal}
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\begin{document}$$C^\infty $$\end{document}-smooth Jordan curve boundary, and if the arc I⊂∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$I\subset \partial \Omega $$\end{document} is nowhere real-analytic, then we have local uniqueness already with sub-Cauchy data: if a function is biharmonic in O∩Ω\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}\cap \Omega $$\end{document} for some planar neighborhood O\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}$$\end{document} of I, and is three-flat on I, then it vanishes identically on O∩Ω\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}\cap \Omega $$\end{document}, provided that O∩Ω\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {O}}\cap \Omega $$\end{document} is connected. Finally, we consider a three-dimensional setting, and analyze it partially using analogues of the square of the standard 2×2\documentclass[12pt]{minimal}
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\begin{document}$$2\times 2$$\end{document} 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.
