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

Grüter, Michael

Arch. Ration. Mech. Anal. 97, 261-270 (1987).

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

Keywords:

partitioning problem; regularity of minimizing varifolds; sets of finite perimeter

Citations:

Zbl 0486.49024

References:

[1]	F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc. 4, no. 165 (1976). · Zbl 0327.49043
[2]	H. Federer, Geometric measure theory. Springer-Verlag, Berlin, Heidelberg, New York (1969). · Zbl 0176.00801
[3]	E. Giusti, Minimal surfaces and functions of bounded variation. Birkhäuser Verlag, Boston, Basel, Stuttgart (1984). · Zbl 0545.49018
[4]	M. Grüter, S. Hildebrandt, & J. C. C. Nitsche, Regularity for stationary surfaces of constant mean curvature with free boundaries. Acta Math. 156 (1986), 119–152. · Zbl 0609.49027
[5]	M. Grüter & J. Jost, Allard type regularity results for varifolds with free boundaries. Ann. d. Sc. Norm. Sup. di Pisa, to appear (1986). · Zbl 0615.49018
[6]	M. Grüter, Regularität von minimierenden Strömen bei einer freien Randbedingung. Habilitationsschrift, Universität Düsseldorf (1985).
[7]	M. Grüter, Regularity results for minimizing currents with a free boundary. Preprint (1985).
[8]	M. Grüter, Optimal regularity for codimension one minimal surfaces with a free boundary. Preprint (1985).
[9]	E. Gonzales, U. Massari, & I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J. 32 (1983), 25–37. · Zbl 0486.49024
[10]	R. Hardt, & L. Simon, Seminar on geometric measure theory. Birkhäuser Verlag, Basel, Boston, Stuttgart (1986). · Zbl 0601.49029
[11]	S. Hildebrandt & W. JÄger, On the regularity of surfaces with prescribed mean curvature at a free boundary. Math. Z. 118 (1970), 289–308. · Zbl 0204.11504
[12]	S. Hildebrandt & H. C. Wente, Variational problems with obstacles and a volume constraint. Math. Z. 135 (1973), 55–68. · Zbl 0284.49017
[13]	C. B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, Berlin, Heidelberg, New York (1966). · Zbl 0142.38701
[14]	L. Simon, Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Canberra, Vol. 3 (1983). · Zbl 0546.49019
[15]	I. Tamanini, Il problema della capillarità su domini non regolari. Rend. Sem. Mat. Univ. Padova 56 (1977), 169–191. · Zbl 0406.76031