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On the nonuniqueness of surfaces with constant mean curvature spanning a given contour. (English) Zbl 0678.49036
The author shows that Rellich’s conjecture holds for any rectifiable closed oriented Jordan curve \(\gamma\) in \(R^ 3\), that is, there exist \(H_*<0\) and \(H^*>0\) such that for any \(H\in (H_*,0)\cup (0,H^*)\) the Plateau problem for generalized surfaces with constant mean curvature H and boundary \(\gamma\) has at least two geometrically distinct solutions. One of these is large in the sense that its area, volume and diameter grow asymptotically as \(H\to 0\), while the other solution is small, i.e. these quantities remain bounded as \(H\to 0.\)
The proof relies on the isoperimetric inequality for the volume functional due to H. Wente and on a nonuniqueness result of M. Struwe [ibid. 93, 135-157 (1986; Zbl 0603.49027)]. The author also shows a similar result for the corresponding Dirichlet problem. Given any nonconstant boundary data he proves the existence of at least two solutions for all sufficiently small \(H\neq 0\), and again one solution is small, the other large.
Added in proof the author mentions the independent and almost simultaneous work of H. Brézis and J.-M. Coron on the same subject [C. R. Acad. Sci., Paris, Sér. I 295, 615-618 (1982; Zbl 0505.49019); Commun. Pure Appl. Math. 37, 149-187 (1984; Zbl 0537.49022)].

MSC:
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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