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Boundary regularity for solutions of a partitioning problem. (English) Zbl 0613.49029

For a bounded smooth domain \(\Omega \subset {\mathbb{R}}^{n+1}\) and \(0<\sigma <1\) the author uses standard arguments coming from the theory of sets of finite perimeter to produce a set \(E\subset \Omega\) satisfying meas E\(=\sigma \cdot \text{meas}\Omega\) such that \(\partial E\cap \Omega\) has minimal area among all such sets. While interior regularity was proved by E. Gonzalez, Z. U. Massari and I. Tamanini [Indiana Univ. Math. J. 32, 25-37 (1983; Zbl 0486.49024)] the author extends their result up to the boundary by showing \({\mathbb{H}}-\dim (\sin g T)\leq n-7\) for the set of singularities in \({\bar \Omega}\) of the associated rectifiable current \(T:=\partial[[E]]L\Omega\). Moreover, near a regular point \(x\in \delta \Omega\) spt T and \(\partial \Omega\) intersect orthogonally, and the varifold associated to T has constant generalized mean curvature. The proofs are based on the author’s earlier work on the regularity for minimal surfaces with a free boundary.
Reviewer: M.Fuchs

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 0486.49024
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References:

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