Polygonal Hele–Shaw problem with surface tension

Abstract

We study a polygonal analogue of the Hele–Shaw moving boundary problem 1 with surface tension based on a framework of polygonal motion proposed by Benesˇ et al. [5]. A key idea is to introduce a polygonal Dirichlet-to-Neumann map. We study variational properties of the polygonal Dirichlet to-Neumann map and show that our polygonal Hele–Shaw problem is a polygonal analogue of the original problem. Local solvability of a polygonal Hele–Shaw problem is also proved by means of the variational structure.