45 research outputs found
On the uniqueness theorem of Holmgren
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 is
the interior domain of an ellipse, and is a proper arc of the ellipse
, then there exists a nontrivial biharmonic function in
which vanishes to degree three on (i.e., all partial derivatives
of of order vanish on ) 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.Comment: 14 page
Hele-Shaw flow on weakly hyperbolic surfaces
We consider the Hele-Shaw flow that arises from injection of two-dimensional
fluid into a point of a curved surface. The resulting fluid domains have and
are more or less determined implicitly by a mean value property for harmonic
functions. We improve on the results of Hedenmalm and Shimorin \cite{HS} and
obtain essentially the same conclusions while imposing a weaker curvature
condition on the surface. Incidentally, the curvature condition is the same as
the one that appears in a recent paper of Hedenmalm and Perdomo, where the
problem of finding smooth area minimizing surfaces for a given curvature form
under a natural normalizing condition was considered. Probably there are deep
reasons behind this coincidence.Comment: 16 page