Let $\Omega \subset R^3$ be a bounded open region. It is well known that there exists a smooth surface $\Gamma$ with minimum area and constant mean curvature which is orthogonal to $\partial \Omega$ and divides $\Omega$ into two regions of equal volume. I wonder if there is a numerical algorithm for constructing such $\Gamma$.
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asked Dec 3, 2014 at 8:41
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$\begingroup$ Do you know if this surface $\Gamma$ satisfies some optimization problem, like, for example, minimizing the surface area? $\endgroup$– Beni BogoselCommented Dec 3, 2014 at 9:28
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$\begingroup$ Yes, I just edited the question. $\endgroup$– A random mathematicianCommented Dec 3, 2014 at 9:35
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1 Answer
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There is an algorithm, based on a $\Gamma$-convergence result, which can find the minimal surface which divides a shape into two regions of equal volume. More details can be found in this link and this other link.
The idea is that instead of looking for a minimal surface, we minimize the sum of the perimeters of the two regions determined by $\Gamma$. Thus the problem is transformed into an optimal partitioning problem. In the link, the examples in $3D$ are only for a division of a cube, but they can be generalized to a general shape, like the ones presented in 2D for the circle and the equilateral triangle. The case needed here is relatively simple, since we search an optimal partition containing only two parts.
Alternatively, you could use the software Evolver, which can treat perimeter optimization problems.
answered Dec 3, 2014 at 9:47