- Source: Almgren regularity theorem
In geometric measure theory, a field of mathematics, the Almgren regularity theorem, proved by Almgren (1983, 2000), states that the singular set of a mass-minimizing surface has codimension at least 2. Almgren's proof of this was 955 pages long. Within the proof many new ideas are introduced, such as monotonicity of a frequency function and the use of a center manifold to perform a more intricate blow-up procedure.
A streamlined and more accessible proof of Almgren's regularity theorem, following the same ideas as Almgren, was given by Camillo De Lellis and Emanuele Spadaro in a series of three papers.
References
Almgren, F. J. (1983), "Q valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two", Bulletin of the American Mathematical Society, New Series, 8 (2): 327–328, doi:10.1090/S0273-0979-1983-15106-6, ISSN 0002-9904, MR 0684900
Almgren, Frederick J. Jr. (2000), Taylor, Jean E.; Scheffer, Vladimir (eds.), 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, vol. 1, River Edge, NJ: World Scientific, ISBN 978-981-02-4108-7, MR 1777737, Zbl 0985.49001
Chang, Sheldon X. (1998), "On Almgren's regularity result", The Journal of Geometric Analysis, 8 (5): 703–708, doi:10.1007/BF02922666, ISSN 1050-6926, MR 1731058, S2CID 120598029
White, Brian (1998), "The mathematics of F. J. Almgren, Jr", The Journal of Geometric Analysis, 8 (5): 681–702, CiteSeerX 10.1.1.120.4639, doi:10.1007/BF02922665, ISSN 1050-6926, MR 1731057, S2CID 122083638
Kata Kunci Pencarian:
- Almgren regularity theorem
- Regularity theorem
- Frederick J. Almgren Jr.
- Almgren–Pitts min-max theory
- List of long mathematical proofs
- Plateau's problem
- Camillo De Lellis
- Richard Schoen
- Ennio De Giorgi
- Bernstein's problem
