Explicit determination of area minimizing hypersurfaces. II. (English) Zbl 0644.53007

Mem. Am. Math. Soc. 342, 90 p. (1986).
The author presents a method for explicitly computing area-minimizing hypersurfaces whose boundaries lie on the surface of a convex body. The basic result is a strengthening of a method previously given by the same author in part I [Duke Math. J. 44, 519–534 (1977; Zbl 0385.49026)]. In distinction from part I, the hypothesis that the area-minimizing hypersurfaces be unique can be dropped here. In lower-dimensional cases additional refinements are made. In particular, the case of a polygonal boundary in \(\mathbb{R}^3\) is very precisely analysed with the aid of a new barrier function appropriate for corners.
The customary notations and terminology in the monograph by H. Federer [Geometric measure theory. [Berlin etc.: Springer-Verlag (1969; Zbl 0176.00801)] will be used here without particular mention. Let \(\Omega\) denote a bounded open convex subset of \(\mathbb{R}^n\) \((n\geq 2)\). For each Lipschitzian map \(u: \Omega\to\mathbb{R}\) the Dirichlet integral \(G[u]:=\int_{\Omega}| Du| \,d{\mathcal L}^n\) is introduced. Now admitting every Lipschitzian map \(u: \Omega\to\mathbb{R}\) with the same boundary correspondence as a prescribed map \(\phi_0\in \operatorname{Lip}(\Omega,\mathbb{R})\) to concurrence, one sets \(G[u_0] = \inf G[u]\). Here it is assumed that \(\phi_0\) satisfies the bounded slope condition with constant \(M\) and that \(\operatorname{Lip}(u_0)\leq M\). So \(u_0\) is of least gradient with respect to \(\Omega\), as in part I. Set \(\Omega_0:=E^n\lfloor \Omega\), \(a:=\inf \{u_0(x): x\in \Omega \}\), \(b:=\sup \{u_0(x): x\in \Omega \}\), and for each \(r\) with \(a<r<b\) set \(T_r:=\partial \Omega_0\lfloor \{x: \phi_0(x)\geq r\}- \partial (\Omega_0\lfloor \{x: u_0(x)\geq r\})\), \(S_r:=\partial (\Omega_0\lfloor \{x: u_0(x)\leq r\})-\partial \Omega_0\lfloor \{x: \phi_0(x)\leq r\}\).
The basic result is the following approximation theorem.
Theorem: Suppose \(a<\alpha <\beta <b\). Then there exists a sequence \(F_1,F_2,\ldots\) of explicitly computable subsets of \(\overline \Omega\) such that
(1) \(\mathcal L^n[F_j]+\mathcal H^{n-1}[\partial \Omega \cap F_j] \leq 2^{- j}(\mathcal L^n[\Omega]+\mathcal H^{n-1}[\partial \Omega]+1)\), \(j=1,2,\ldots\);
(2) for some \(r\), with \(\alpha <r<\beta\), (i) \(\cap_{j=1,2,\ldots}F_j=\overline \Omega\cap u_0^{-1}(r)\), and (ii) \(\Omega\cap \cap_{j=1,2,\ldots}F_j=\Omega \cap \text{spt }T_r=\Omega \cap \text{spt }S_r\).
Reviewer: Harold R. Parks


53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
49Q05 Minimal surfaces and optimization
49Q20 Variational problems in a geometric measure-theoretic setting
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