On the uniqueness theorem of Holmgren

Abstract

We rereview 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 vanishes to degree three on $I$ (i.e., all partial derivatives of $u$ of order $\le2$ vanish on $I$) if and only if the ellipse is a circle. Finally, we consider a three-dimensional case, and analyze it partially using analogues of the square of the 2X2 Cauchy-Riemann operator.