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Area minimizing hypersurfaces with prescribed volume and boundary. (English) Zbl 0787.53056
The authors treat “Rellich’s conjecture” in higher dimension which originally is the non-uniqueness of hypersurfaces of prescribed mean curvature with given boundary. This conjecture states that for a given \((n-1)\)-dimensional boundary \(B\) in \(\mathbb{R}^{n+1}\) there are numbers \(c_ - < 0 < c_ +\) such that for every \(H \in [c_ -,c_ +]\setminus \{0\}\) there exist at least two different \(n\)-dimensional surfaces with boundary \(B\) and constant mean curvature \(H\). One small and one large solution in analogy to spherical caps with the same boundary and the same \(H\). This problem has been solved by Struwe and Brezis/Coron for two- dimensional parametric surfaces in \(\mathbb{R}^ 3\). To treat the higher dimensional situation the authors work in the setting of geometric measure theory and study the corresponding volume constrained problem for locally rectifiable integer multiplicity \(n\)-currents in \(\mathbb{R}^{n+1}\).
Besides many results for existence and regularity and asymptotic behaviour for large volumes for minimizers, one of the main results is the following theorem: There exist infinitely many values \(H > 0\) and infinitely many values \(H < 0\) accumulating at 0 such that one has at least two indecomposable \(n\)-currents with constant mean curvature \(H\) and boundary \(B\) with certain regularity properties.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q20 Variational problems in a geometric measure-theoretic setting
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[1] [All 1] Allard, W.K.: On the first variation of a varifold. Ann. Math.95, 417–491 (1972) · Zbl 0252.49028
[2] [All 2] Allard, W.K.: On the first variation of a varifold: boundary behaviour. Ann. Math.101, 418–446 (1975) · Zbl 0319.49026
[3] [Alm 1] Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Am. Math. Soc.165 (1976) · Zbl 0327.49043
[4] [Alm 2] Almgren, F.J.: Optimal Isoperimetric Inequalities: Indiana Univ. Math. J.35, 451–547 (1986) · Zbl 0597.49029
[5] [BC] Brezis, H., Coron, J.-M.: Multiple solutions of H-systems and Rellich’s conjecture. Commun. Pure Appl. Math.37, 149–187 (1984) · Zbl 0537.49022
[6] [DF 1] Duzaar, F., Fuchs, M.: On integral currents with constant mean curvature. Rend. Semin. Mat. Univ. Padova85, 79–103 (1991) · Zbl 0739.49026
[7] [DF2] Duzaar, F., Fuchs, M.: A general existence theorem for integral currents with prescribed mean curvature form. Boll. Unione Mat. Ital. (to appear) · Zbl 0786.49025
[8] [DF 3] Duzaar, F., Fuchs, M.: On the existence of integral currents with prescribed mean curvature vector. Manuscr. Math.67, 41–67 (1990) · Zbl 0703.49035
[9] [DS] Duzaar, F., Steffen, K.: A partial regularity theorem for harmonic maps at a free boundary. Asymptotic Anal.2, 299–343 (1989) · Zbl 0699.58025
[10] [Fe] Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969 · Zbl 0176.00801
[11] [Fe 1] Federer, H.: The singular set of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Am. Math. Soc.76, 767–771 (1979) · Zbl 0194.35803
[12] [He] Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Ration. Mech. Anal.35, 249–252 (1969) · Zbl 0184.32802
[13] [HS] Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Ann. Math.110, 439–486 (1979) · Zbl 0457.49029
[14] [Gi] Giusti, E.: Minmal surfaces and functions of bounded variation. (Monogr. Math., Basel) Basel Boston Stuttgart: Birkhäuser 1984
[15] [GG] Gonzalez, E.H.A., Greco, G.H.: Una nuova dimonstrazione della proprietà isoperimetrica dell’ipersfera nella classe degli insiemi aventi perimetro finito. Ann. Univ. Ferrara23, 251–256 (1977) · Zbl 0375.28016
[16] [GMT] Gonzalez, E., Massari, U., Tamanini, I.: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J.32, 25–37 (1983) · Zbl 0504.49026
[17] [GT] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, second edition. Berlin Heidelberg New York: Springer 1977 · Zbl 0361.35003
[18] [Gu] Gulliver, R.: On the nonexistence of a hypersurface of prescribed mean curvature with given boundary. Manuscr. Math.11, 15–39 (1974) · Zbl 0266.53002
[19] [MM] Massari, U., Miranda, M.: Minimal surfaces of codimension one. (Math. Stud., vol. 91) Amsterdam: North Holland 1984 · Zbl 0565.49030
[20] [Mo] Morrey, C.B.: Second order elliptic systems of differential equations (Ann. Math. Stud., vol. 33, pp. 101–159) Princeton: Princeton University Press 1954 · Zbl 0057.08301
[21] [Si] Simon, L.: Lectures on geometric measure theory: Canberra: Aust. Natl. Univ. 1984 (Proc. Cent. Math. Anal. Aust. Natl. Univ., vol. 3)
[22] [Ste 1] Steffen, K.: Flächen vorgeschriebener mittlerer Krümmung mit vorgegebenem Volumen oder Flächeninhalt. Arch. Ration. Mech. Anal.49, 191–217 (1972) · Zbl 0259.53043
[23] [Ste 2] Steffen, K.: On the nonuniqueness of surfaces with prescribed mean curvature spanning a given contour. Arch. Ration. Mech. Anal.94, 101–122 (1986) · Zbl 0678.49036
[24] [Str 1] Struwe, M.: Nonuniqueness in the Plateau-problem for surfaces of constant mean curvature. Arch. Ration. Mech. Anal.93, 135–157 (1986) · Zbl 0603.49027
[25] [Str 2] Struwe, M.: Large H-surfaces via the mountain-pass-lemma. Math. Ann.270, 441–459 (1985) · Zbl 0582.58010
[26] [We 1] Wente, H.: A general existence theorem for surfaces of constant mean curvature. Math. Z.120, 277–288 (1971) · Zbl 0214.11101
[27] [We 2] Wente, H.: The Dirichlet problem with a volume constraint. Manuscr. Mat.11, 141–157 (1974) · Zbl 0268.35031
[28] [We 3] Wente, H.: Large solutions to the volume constrained Plateau problem. Arch. Ration. Mech. Anal.75, 59–77 (1980) · Zbl 0473.49029
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