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. . 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.
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Masato Kimura, Daisuke Tagami, Shigetoshi Yazaki, Polygonal Hele–Shaw problem with surface tension. Interfaces Free Bound. 15 (2013), no. 1, pp. 77–93DOI 10.4171/IFB/295