Variational Methods in Potential Theory and Planar Elliptic Growth.

摘要

A nested family of growing or shrinking planar domains is called a Laplacian growth process if the normal velocity of each domain's boundary is propor- tional to the gradient of the domain's Green function with a fixed singularity on the interior. In this dissertation we consider a generalization to so-called elliptic growth, wherein the Green function is replaced with that of a more general elliptic operator, which models inhomogeneities in the underlying plane. Of particular interest is the way that elliptic growth extends Laplacian growth. As such, we consider elliptic operators that are somehow close to the Laplacian and derive perturbative formulas for the Green function; with these we discuss a couple of inverse problems which seek to locally characterize the newly enlarged phase space.