December  2011, 31(4): 1249-1272. doi: 10.3934/dcds.2011.31.1249

Center manifold: A case study

1. 

Universität Zürich, Institut für Mathematik, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland

2. 

Hausdorff Center for Mathematics, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Received  February 2010 Revised  January 2011 Published  September 2011

Following Almgren's construction of the center manifold in his Big regularity paper, we show the $C^{3,\alpha}$ regularity of area-minimizing currents in the neighborhood of points of density one without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren's construction in its full generality.
Citation: Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
References:
[1]

Frederick J. Almgren, Jr., "Almgren's Big Regularity Paper. $Q$-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2," World Scientific Monograph Series in Mathematics, 1,, World Scientific Publishing Co., (2000).   Google Scholar

[2]

Ennio De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa 1960-61, (1961), 1960.   Google Scholar

[3]

Camillo De Lellis and Emanuele Nunzio Spadaro, Higher integrability and approximation of minimal currents,, preprint, ().   Google Scholar

[4]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).   Google Scholar

[5]

Eberhard Hopf, Über den funktionalen, insbesondere den analytischen Charakter der Lösun-gen elliptischer Differentialgleichungen zweiter Ordnung,, Math. Z., 34 (1932), 194.  doi: 10.1007/BF01180586.  Google Scholar

[6]

Leon Simon, "Lectures on Geometric Measure Theory,", Proceedings of the Centre for Mathematical Analysis, 3 (1983).   Google Scholar

[7]

Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43,, Monographs in Harmonic Analysis, (1993).   Google Scholar

show all references

References:
[1]

Frederick J. Almgren, Jr., "Almgren's Big Regularity Paper. $Q$-Valued Functions Minimizing Dirichlet's Integral and the Regularity of Area-Minimizing Rectifiable Currents up to Codimension 2," World Scientific Monograph Series in Mathematics, 1,, World Scientific Publishing Co., (2000).   Google Scholar

[2]

Ennio De Giorgi, "Frontiere Orientate di Misura Minima,", Seminario di Matematica della Scuola Normale Superiore di Pisa 1960-61, (1961), 1960.   Google Scholar

[3]

Camillo De Lellis and Emanuele Nunzio Spadaro, Higher integrability and approximation of minimal currents,, preprint, ().   Google Scholar

[4]

Lawrence C. Evans and Ronald F. Gariepy, "Measure Theory and Fine Properties of Functions,", Studies in Advanced Mathematics, (1992).   Google Scholar

[5]

Eberhard Hopf, Über den funktionalen, insbesondere den analytischen Charakter der Lösun-gen elliptischer Differentialgleichungen zweiter Ordnung,, Math. Z., 34 (1932), 194.  doi: 10.1007/BF01180586.  Google Scholar

[6]

Leon Simon, "Lectures on Geometric Measure Theory,", Proceedings of the Centre for Mathematical Analysis, 3 (1983).   Google Scholar

[7]

Elias M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43,, Monographs in Harmonic Analysis, (1993).   Google Scholar

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