Isoperimetric Problems Between Analysis and Geometry

Improved convergence theorem for perimeter minimizing clusters

Francesco Maggi (UT Austin)

created by paolini on 02 Jun 2014
modified on 03 Jun 2014

19 jun 2014 -- 11:00   [open in google calendar]

Abstract.

A well-known fact about sequences of perimeter almost-minimizing sets is that $L^1$ convergence (convergence to zero of the volume of the symmetric difference) improves to $C^1$ convergence (existence of boundary diffeomorphisms which converge to the identity map in $C^1$) whenever the limit set has smooth boundary. This is a classical application of the small excess regularity criterion, which is useful in showing the equivalence of $L^1$-local and $C^1$-local minimality conditions, as well as in proving quantitative stability inequalities, and in providing qualitative descriptions of minimizers in surface tension driven problems. The smoothness assumption on the limit set is automatically valid in dimension less or equal than 7, but may fail otherwise. Our understanding of these singularities is to lacunary to allow for an extension of the above ”improved convergence theorem” when the limit set is singular. When one moves from the framework of sets to that of clusters (modeling, say, soap bubble compounds) singularities appear even in dimension 2. However, in the case of clusters, we have a very good understanding of singularities in dimension 2 and 3, based on Jean Taylor’s theorem on the validity of Plateau’s laws. Starting from this sharp local description of bubble clusters we explain how to prove improved convergence theorems for sequences of singular sets of perimeter almost-minimizing bubble clusters in $\mathbb R^2$ and $\mathbb R^3$, and briefly discuss some of the possible applications of these results.

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