Complex Analysis and Operator Theory
Dmitry Khavinson, Mark B. Mineev-Weinstein, Mihai Putinar · 2009
The planar elliptic extension of the Laplacian growth is, after a proper parametrization, given in a form of a solution to the equation for area- preserving diffeomorphisms. The infinite set of conservation laws associated with such elliptic growth is interpreted in terms of potential theory, and the relations between two major forms of the elliptic growth are analyzed. The constants of integration for closed form solutions are identified as the singu- larities of the Schwarz function, which are located both inside and outside the moving contour. Well-posedness of the recovery of the elliptic operator gov- erning the process from the continuum of interfaces parametrized by time is addressed and two examples of exact solutions of elliptic growth are presented. Mathematics Subject Classification (2000). Primary 76S05 ; Secondary 76D27,