## 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

### MSC:

 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

### Citations:

Zbl 0385.49026; Zbl 0176.00801
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