Home Existence and soap film regularity of solutions to Plateau’s problem
Article
Licensed
Unlicensed Requires Authentication

Existence and soap film regularity of solutions to Plateau’s problem

  • Jenny Harrison EMAIL logo and Harrison Pugh
Published/Copyright: August 29, 2015

Abstract

Plateau’s problem is to find a surface with minimal area spanning a given boundary. Our paper presents a theorem for codimension one surfaces in n in which the usual homological definition of span is replaced with a novel algebraic-topological notion. In particular, our new definition offers a significant improvement over existing homological definitions in the case that the boundary has multiple connected components. Let M be a connected, oriented compact manifold of dimension n-2 and 𝔖 the collection of compact sets spanning M. Using Hausdorff spherical measure as a notion of “size,” we prove: There exists an X0 in 𝔖 with smallest size. Any such X0 contains a “core” X0*𝔖 with the following properties: It is a subset of the convex hull of M and is a.e. (in the sense of (n-1)-dimensional Hausdorff measure) a real analytic (n-1)-dimensional minimal submanifold. If n=3, then X0* has the local structure of a soap film. Furthermore, set theoretic solutions are elevated to current solutions in a space with a rich continuous operator algebra.

MSC 2010: 49J45; 46G99; 46T30

Communicated by Frank Duzaar


References

[1] Allard W. K., On the first variation of a varifold, Ann. of Math. (2) 95 (1972), no. 3, 417–491. 10.2307/1970868Search in Google Scholar

[2] Almgren F. J., Theory of Varifolds, mimeographed notes, 1965. Search in Google Scholar

[3] Almgren F. J., Plateau’s Problem. An Invitation to Varifold Geometry, Benjamin, New York, 1966. Search in Google Scholar

[4] Almgren F. J., Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. (2) 87 (1968), no. 2, 321–391. 10.2307/1970587Search in Google Scholar

[5] Almgren F. J., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Bull. Amer. Math. Soc. 81 (1975), no. 1, 151–155. 10.1090/S0002-9904-1975-13681-0Search in Google Scholar

[6] Almgren F. J., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), No. 165. Search in Google Scholar

[7] Besicovitch A. S., Parametric surfaces III: On surfaces of minimum area, J. Lond. Math. Soc. 23 (1948), 241–246. 10.1112/jlms/s1-23.4.241Search in Google Scholar

[8] Besicovitch A. S., Parametric surfaces I: Compactness, Proc. Cambridge Phil. Soc. 45 (1949), 1–13. 10.1017/S0305004100000396Search in Google Scholar

[9] Bombieri E., Recent progress in the theory of minimal surfaces, Enseign. Math. 25 (1979), 1–9. Search in Google Scholar

[10] Caccioppoli R., Sulla quadratura delle superfici piane e curve, Atti Accad. Naz. Lincei Rend. Lincei Sci. Fuis. Mat. Nat. 6 (1927), 142–146. Search in Google Scholar

[11] Courant R., Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces, Interscience, New York, 1950. Search in Google Scholar

[12] Dal Maso G., Morel J. M. and Solimini S., A variational method in image segmentation: Existence and approximation results, Acta Math. 168 (1992), no. 1, 89–151. 10.1007/BF02392977Search in Google Scholar

[13] David G., Limits of Almgren quasiminimal sets, Harmonic Analysis at Mount Holyoke (South Hadley 2001), Contemp. Math. 320, American Mathematical Society, Providence (2003), 119–145. 10.1090/conm/320/05603Search in Google Scholar

[14] David G., Local regularity properties of almost- and quasiminimal sets with a sliding boundary condition, preprint 2014, http://arxiv.org/abs/1401.1179. 10.24033/ast.1077Search in Google Scholar

[15] David G. and Semmes S., Uniform rectifiability and quasiminimizing sets of arbitrary codimension, Mem. Amer. Math. Soc. 144 (2000), No. 687. 10.1090/memo/0687Search in Google Scholar

[16] De Giorgi D. E., Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa (1960–1961), Editrice Tecnico Scientifica, Pisa (1961), 1–56. Search in Google Scholar

[17] De Lellis C., Allard’s interior regularity theorem: An invitation to stationary varifolds, Lecture notes, 2012, http://www.math.uzh.ch/fileadmin/user/delellis/publikation/allard_31.pdf. Search in Google Scholar

[18] De Lellis C., Ghiraldin F. and Maggi F., A direct approach to Plateau’s problem, preprint 2014, http://arxiv.org/abs/1408.4047. 10.4171/JEMS/716Search in Google Scholar

[19] de Pauw T., Size minimizing surfaces with integral coefficients, Ann. Sci. Ec. Norm. Supér. (4) 42 (2009), no. 1, 37–101. 10.24033/asens.2090Search in Google Scholar

[20] De Philippis G., De Rosa A. and Ghiraldin F., A direct approach to plateau’s problem in any codimension, preprint 2015, http://arxiv.org/abs/1501.07109. 10.1016/j.aim.2015.10.007Search in Google Scholar

[21] Douglas J., Solutions of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), 263–321. 10.1090/S0002-9947-1931-1501590-9Search in Google Scholar

[22] Federer H., Geometric Measure Theory, Springer, Berlin, 1969. Search in Google Scholar

[23] Federer H. and Fleming W. H., Normal and integral currents, Ann. of Math. (2) 72 (1960), no. 3, 458–520. 10.2307/1970227Search in Google Scholar

[24] Folland G., Real Analysis. Modern Techniques and Their Applications, 2nd ed., John Wiley and Sons, New York, 1999. Search in Google Scholar

[25] Harrison J., Stokes’ theorem on nonsmooth chains, Bull. Amer. Math. Soc. 29 (1993), 235–242. 10.1090/S0273-0979-1993-00429-4Search in Google Scholar

[26] Harrison J., On Plateau’s problem for soap films with a bound on energy, J. Geom. Anal. 14 (2004), no. 2, 319–329. 10.1007/BF02922075Search in Google Scholar

[27] Harrison J., Ravello lecture notes, 2005. Search in Google Scholar

[28] Harrison J., Operator calculus – the exterior differential complex, preprint 2011, http://arxiv.org/abs/1101.0979. Search in Google Scholar

[29] Harrison J., Soap film solutions to Plateau’s problem, preprint 2012, http://arxiv.org/abs/1106.5839. 10.1007/s12220-012-9337-xSearch in Google Scholar

[30] Harrison J., Operator calculus of differential chains and differential forms, J. Geom. Anal. 24 (2014), no. 1, 271–297. 10.1007/s12220-013-9433-6Search in Google Scholar

[31] Harrison J., Soap film solutions to Plateau’s problem, J. Geom. Anal. 24 (2014), 271–297. 10.1007/s12220-012-9337-xSearch in Google Scholar

[32] Harrison J. and Pugh H., Topological aspects of differential chains, J. Geom. Anal. 22 (2012), no. 3, 685–690. 10.1007/s12220-010-9210-8Search in Google Scholar

[33] Harrison J. and Pugh H., Existence and soap film regularity of solutions to Plateau’s problem, preprint 2013, http://arxiv.org/abs/1310.0508. 10.1515/acv-2015-0023Search in Google Scholar

[34] Harrison J. and Pugh H., Spanning via cech cohomology, preprint 2014, http://arxiv.org/abs/1412.2193. Search in Google Scholar

[35] Harrison J. and Pugh H., Solutions to Lipschitz variational problems with cohomological spanning conditions, preprint 2015, http://arxiv.org/abs/1506.01692. Search in Google Scholar

[36] Hirsch M. W., Differential Topology, Springer, Berlin, 1976. 10.1007/978-1-4684-9449-5Search in Google Scholar

[37] Lagrange J.-L., Essai d’une nouvelle méthode pour détérminer les maxima et les minima des formules intégrales indéfinies, Miscellanea Taurinensia 2 (1760–1761), 335–362. Search in Google Scholar

[38] Lebesgue H., Intégrale, longueur, aire, Annali di Mat. (3) 7 (1902), 231–359. 10.1007/BF02420592Search in Google Scholar

[39] Maggi F., Sets of Finite Perimeter and Geometric Variational Problems, Cambridge University Press, Cambridge, 2012. 10.1017/CBO9781139108133Search in Google Scholar

[40] Massey W., On the normal bundle of a sphere imbedded in Euclidean space, Proc. Amer. Math. Soc. 10 (1959), no. 6, 959–964. 10.1090/S0002-9939-1959-0109351-8Search in Google Scholar

[41] Mattila P., Geometry of sets And Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge University Press, Cambridge, 1999. Search in Google Scholar

[42] Pepling R., Soap bubbles, Chem. Eng. News 81 (2003), No. 17. Search in Google Scholar

[43] Plateau J., Experimental and Theoretical Statics of Liquids Subject to Molecular Forces Only, Gauthier-Villars, Paris, 1873. Search in Google Scholar

[44] Podio-Guidugli P., A primer in elasticity, J. Elasticity 58 (2000), 1–104. 10.1007/978-94-017-0594-3Search in Google Scholar

[45] Pugh H., Applications of differential chains to complex analysis and dynamics, Harvard senior thesis, Department of Mathematics, Harvard University, 2009; http://arxiv.org/abs/1012.5542. Search in Google Scholar

[46] Radó T., On the Problem of Plateau, Ergeb. Math. Grenzgeb. (2) 2, Julius Springer, Berlin, 1933. 10.1007/978-3-642-99118-9Search in Google Scholar

[47] Reifenberg E. R., Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), no. 1–2, 1–94. 10.1007/BF02547186Search in Google Scholar

[48] Reifenberg E. R., Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Bull. Amer. Math. Soc. 66 (1960), no. 4, 312–313. 10.1090/S0002-9904-1960-10482-XSearch in Google Scholar

[49] Taylor J., The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 2, 489–539. 10.2307/1970949Search in Google Scholar

[50] Ward D. J., A counterexample in area theory, Math. Proc. Cambridge Philos. Soc. 60 (1964), no. 4, 821–845. 10.1017/S0305004100038317Search in Google Scholar

[51] Whitney H., Geometric Integration Theory, Princeton University Press, Princeton, 1957. 10.1515/9781400877577Search in Google Scholar

Received: 2015-5-29
Revised: 2015-7-13
Accepted: 2015-7-16
Published Online: 2015-8-29
Published in Print: 2016-10-1

© 2016 by De Gruyter

Downloaded on 26.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/acv-2015-0023/html
Scroll to top button