Evolution of spoon-shaped networks

Abstract

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain \(\Omega\). The two curves meet only in one point, forming angle of \(120\) degrees. The non-closed curve has a fixed end point on \(\partial\Omega\). We study the evolution by curvature of this network. We show that the maximal existence time depends only on the area enclosed in the initial loop, if the length of the non-closed curve stays bounded from below during the evolution. Moreover, the closed curve shrinks to a point and the network is asymptotically approaching, after dilations and extraction of a subsequence, a Brakke spoon.