## 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

Zbl 0486.49024
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### References:

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