Abstract
We prove some facts concerning surfaces of minimal area bounding regions of prescribed volume in ℝn. The main result we prove is that the mean curvature of such a surface is constant, if possibly a discontinuous function of the enclosed volume. The boundary behaviour of the solutions is also discussed.
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ANZELLOTTI, G., GIAQUINTA, M., MASSARI, U., MODICA, G., PEPE, L.: Note sul problema di Plateau. Pisa: Editrice Tecnico Scientifica 1974
CHEN, J.T.: On the existence of capillary free surfaces in the absence of gravity. Doctoral dissertation. Stanford University (1979)
CONCUS, P., FINN, R.: On capillary free surfaces in the absence of gravity. Acta Math.132, 177–198 (1974)
DE GIORGI, E., COLOMBINI, F., PICCININI, L.: Frontiere orientate di misura minima e questione collegate. Pisa: Editrice Tecnico Scientifica 1972
FINN, R.: Existence and non existence of capillary surfaces. Manuscripta Math.28, 1–11 (1979)
FINN, R., GIUSTI, E.: Non existence and existence of capillary surfaces. Manuscripta Math.28, 13–20 (1979)
GERHARDT, C.: On the capillarity problem with constant volume. Ann. Scuola Norm. Sup. Pisa (4)2, 303–320 (1975)
GIAQUINTA, M.: Regolarità delle superfici BV(Ω) con curvatura media assegnata. Boll. U.M.I.8, 567–578 (1973)
GIAQUINTA, M.: On the Dirichlet problem for surfaces of prescribed mean curvature. Manuscripta Math.12, 73–86 (1974)
GIUSTI, E.: Generalized solutions for the mean curvature equation. To appear
GIUSTI, E.: The equilibrium configuration of liquid drops. To appear in J. für Reine und Angew. Math.
GONZALEZ, E., MASSARI, U., TAMANINI, I.: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. To appear
MASSARI, U.: Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in ℝn. Arch. Rat. Mech. Anal.55, 357–382 (1974)
MASSARI, U., PEPE, L.: Successioni convergenti di ipersuperfici di curvatura media assegnata. Rend. Sem. Mat. Univ. Padova53, 53–68 (1975)
MIRANDA, M.: Existence and regularity of hypersurfaces of ℝn with prescribed mean curvature. Proceedings of Symposia in Pure Mathematics,23 (1973)
MIRANDA, M.: Frontiere minimali con ostacoli. Ann. Univ. Ferrara16, 29–37 (1971)
SERRIN, J.: The problem of Dirichlet for quasilinear elliptic equations with many independent variables. Phil. Trans. Royal Soc. London264, 413–496 (1969)
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Gonzalez, E., Massari, U. & Tamanini, I. Minimal boundaries enclosing a given volume. Manuscripta Math 34, 381–395 (1981). https://doi.org/10.1007/BF01165546
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DOI: https://doi.org/10.1007/BF01165546