4 research outputs found
Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential
The Schwarz function has played an elegant role in understanding and in
generating new examples of exact solutions to the Laplacian growth (or "Hele-
Shaw") problem in the plane. The guiding principle in this connection is the
fact that "non-physical" singularities in the "oil domain" of the Schwarz
function are stationary, and the "physical" singularities obey simple dynamics.
We give an elementary proof that the same holds in any number of dimensions for
the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17]
(1989). A generalization is also given for the so-called "elliptic growth"
problem by defining a generalized Schwarz potential. New exact solutions are
constructed, and we solve inverse problems of describing the driving
singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n -
techniques can be used to locate the singularity set of the Schwarz potential.
One of our methods is to prolong available local extension theorems by
constructing "globalizing families". We make three conjectures in potential
theory relating to our investigation
Random Matrices in 2D, Laplacian Growth and Operator Theory
Since it was first applied to the study of nuclear interactions by Wigner and
Dyson, almost 60 years ago, Random Matrix Theory (RMT) has developed into a
field of its own within applied mathematics, and is now essential to many parts
of theoretical physics, from condensed matter to high energy. The fundamental
results obtained so far rely mostly on the theory of random matrices in one
dimension (the dimensionality of the spectrum, or equilibrium probability
density). In the last few years, this theory has been extended to the case
where the spectrum is two-dimensional, or even fractal, with dimensions between
1 and 2. In this article, we review these recent developments and indicate some
physical problems where the theory can be applied.Comment: 88 pages, 8 figure