On the uniqueness theorem of Holmgren

Abstract

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