zbMATH — the first resource for mathematics

On the nonexistence of a hypersurface of prescribed mean curvature with a given boundary. (English) Zbl 0266.53002

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
49Q05 Minimal surfaces and optimization
35J67 Boundary values of solutions to elliptic equations and elliptic systems
53C40 Global submanifolds
Full Text: DOI EuDML
[1] Bernstein, S.: Sur les surfaces définies au moyen du leur courbure moyenne ou totale. Ann. Ec. Norm. Sup. (Ser. 3)27, 233-256 (1909)
[2] Chern, S.S.: On the curvature of a piece of hypersurface in euclidean space. Abh. Math. Sem. Univ. Hamburg29, 77-91 (1965) · Zbl 0147.20901
[3] Gulliver, R.: Regularity of minimizing surfaces of prescribed mean curvature. To appear in Annals of Math. · Zbl 0235.53042
[4] Gulliver, R.: The Plateau problem for surfaces of prescribed mean curvature in a riemannian manifold. To appear in J. Differential Geometry · Zbl 0275.53033
[5] Gulliver, R., Lesley, F. D.: On boundary branch points of minimizing surfaces. To appear in Arch. Rat. Mech. Anal. · Zbl 0263.53009
[6] Gulliver, R., Osserman, R., Royden, H. L.: A theory of branched immersions of surfaces. Preprint · Zbl 0295.53002
[7] Gulliver, R., Spruck, J.: The Plateau problem for surfaces of prescribed mean curvature in a cylinder. Inventiones Math. 13, 169-178 (1971) · Zbl 0214.11103
[8] Gulliver, R., Spruck, J.: Existence theorems for parametric surfaces of prescribed mean curvature. Indiana U. Math. J.22, 445-472 (1972) · Zbl 0233.53004
[9] Heinz, E.: Über Flächen mit eineindeutiger Projektion auf eine Ebene, deren Krümmungen durch Ungleichungen eingeschränkt sind. Math. Ann.129, 451-454 (1955) · Zbl 0065.37201
[10] Heinz, E.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Rat. Mech. Anal.35, 249-252 (1969) · Zbl 0184.32802
[11] Heinz, E.: Über das Randverhalten quasilinearer elliptischer Systeme mit isothermen Parametern. Math. Z.113, 99-105 (1970) · Zbl 0181.11404
[12] Hildebrandt, S.: Randwertprobleme für Flächen mit vorgeschriebener mittlerer Krümmung und Anwendungen auf die Kapillaritätstheorie I. Math. Z.112, 205-213 (1969) · Zbl 0177.15104
[13] Hildebrandt, S., Kaul, H.: Two-dimensional variational problems with obstructions, and Plateau’s problem for H-surfaces in a Riemannian manifold. Comm. Pure Appl. Math.25, 187-223 (1972) · Zbl 0245.53006
[14] Katsurada, Y.: On a piece of hypersurface in a Riemannian manifold with mean curvature bounded away from zero. Trans. Amer. Math. Soc.129, 447-457 · Zbl 0172.46804
[15] Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Phil. Trans. Royal Soc. London (Ser.A)264, 413-496 (1969) · Zbl 0181.38003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.