Acta Mathematica

Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals

R. Schoen, L. Simon, and F. J. Almgren

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Note

This research was supported in part by grants from the National Science Foundation. Part of the work of the second author was carried out at the Courant Institute of Mathematical Sciences and was supported by a grant from the Alfred P. Sloan Foundation. Part of the work of the third author was supported by a grant from the John Simon Guggenheim Foundation.

Article information

Source
Acta Math., Volume 139 (1977), 217-265.

Dates
Received: 1 September 1976
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485889969

Digital Object Identifier
doi:10.1007/BF02392238

Mathematical Reviews number (MathSciNet)
MR467476

Zentralblatt MATH identifier
0386.49030

Rights
1977 © Almqvist & Wiksell

Citation

Schoen, R.; Simon, L.; Almgren, F. J. Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. Acta Math. 139 (1977), 217--265. doi:10.1007/BF02392238. https://projecteuclid.org/euclid.acta/1485889969


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References for Part I and Part II

  • Almgren, F.J., Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. of Math., 87 (1968), 321–391.
  • Almgren, F. J., Jr., Existence and regularity almost overywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc., 165 (1976).
  • Almgren, F. J., Jr. & Thurston, W. P., Examples of unknotted curves which bound only surfaces of high genus within their convex hulls. Ann. of Math., 105 (1977), 527–538.
  • Allard, W. K., On the first variation of a varifold. Ann. of Math., 95 (1972), 417–491.
  • Bombieri, E., De Giorgi, E., Miranda, M., Una maggiorazione a priori relativa alle impersuperfiei minimali non parametriche. Arch. Rational Mech. Anal., 32 (1969), 255–267.
  • De Giorgi, E., Frontiere orientate di misura minima, Seminario di Mat. della Scuola Normale Superiore, Pisa (1960–61).
  • Federer, H., Geometric Measure Theory. Springer-Verlag New York, 1969.
  • —, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc., 76 (1970), 767–771.
  • Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Berlin, Sber. Preuss, Akad. Wiss., 19 (1927), 147–152.
  • Hardt, R. M., On boundary regularity for integral currents and flat chains modulo 2 minimizing the integral of an elliptic integrand. Preprint.
  • Ladyzhenskaya, O. A. & Ural'tseva, N. N., Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Comm. Pure Appl. Math., 23 (1970), 677–703.
  • Morrey, C. B., Jr., Multiple Integrals in the Calculus of Variations. Springer-Verlag, New York, 1966.
  • Miranda, M. Sulle singolarità della frontiere minimali. Rend. Sem. Mat. Padova, (1967), 181–188.
  • Serrin, J., The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. Roy. Soc. London. Ser. A, 264A (1969), 413–496.
  • —, On the strong maximum principle for quasilinear second order differential inequalities. J. Funct. Anal., 5 (1970), 184–193.
  • Simon, L., Interior gradient bounds for non-uniformly elliptic equations. Indiana Univ. Math. J., 25 (1976), 821–855.
  • —, Remarks on curvature estimates. Duke Math. J., 43 (1976), 545–553.
  • Schoen, R. & Simon, L., A new proof of the regularity theorem for currents minimizing parametric elliptic functionals. To appear.