It was recently noticed that lemniscates do not survive Laplacian growth (Khavinson et al. in Math Res Lett 17:335–341, 2010). This raises the question: “Is there a growth process for which polynomial lemniscates are solutions?” The answer is “yes”, and the law governing the boundary velocity is reciprocal to that of Laplacian growth. Viewing lemniscates as solutions to a moving-boundary problem gives a new perspective on results from classical potential theory, and interesting properties emerge while comparing lemniscate growth to Laplacian growth.
Analysis and Mathematical Physics – Springer Journals
Published: Sep 30, 2012