The double-bubble problem on the square lattice

Abstract

We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of minimisers and, in some parameter regime, we compute the optimal perimeter as a function of the size of the point sets. Moreover, we provide a sharp bound on the difference between two minimisers, which are generally not unique, and use it to rigorously identify their Wulff shape, as the size of the point sets scales up.

Discrete $\ell^{1}$ Double Bubble solution is at most ceiling +2 of the continuous solution

“…In this paper we prove that the solution of the perimeter of the discrete Double bubble solution on the square lattice is at most the ceiling plus two of the continuous solution, where the perimeter is taken with respect to the 1 norm. While finishing this paper, Friedrich, Górny, and Stefanelli [8] have uploaded a paper with mutually exclusive but very related results -mainly showing that different solutions of the discrete Double Bubble problem are close to each other in a geometric sense.…”