July  2022, 42(7): 3233-3299. doi: 10.3934/dcds.2022015

Regularity of almost-minimizers of Hölder-coefficient surface energies

University of Washington, Department of Mathematics, Box 354350, C-138 Padelford, Seattle, WA 98195-4350, USA

Received  June 2021 Published  July 2022 Early access  March 2022

Fund Project: The author was partially supported by NSF FRG 1853993

We study almost-minimizers of anisotropic surface energies defined by a Hölder continuous matrix of coefficients acting on the unit normal direction to the surface. In this generalization of the Plateau problem, we prove almost-minimizers are locally Hölder continuously differentiable at regular points and give dimension estimates for the size of the singular set. We work in the framework of sets of locally finite perimeter and our proof follows an excess-decay type argument.

Citation: David A. Simmons. Regularity of almost-minimizers of Hölder-coefficient surface energies. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3233-3299. doi: 10.3934/dcds.2022015
References:
[1]

W. K. Allard, On the first variation of a varifold, Ann. of Math., 95 (1972), 417-491.  doi: 10.2307/1970868.

[2]

W. K. Allard, A characterization of the area integrand, Symposia Mathematica, 14 (1974), 429-444. 

[3]

F. J. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math. (2), 84 (1966), 277-292.  doi: 10.2307/1970520.

[4]

F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Annals of Mathematics, 87 (1968), 321-391.  doi: 10.2307/1970587.

[5]

E. Bombieri, Regularity theory for almost minimal currents, Arch. Rational Mech. Anal., 78 (1982), 99-130.  doi: 10.1007/BF00250836.

[6]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Ivent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[7]

S. Bortz, M. Engelstein, M. Goering, T. Toro and Z. Zhao, Two phase free boundary problem for Poisson kernels, 2019.

[8]

G. DavidM. EngelsteinM. Smit Vega Garcia and T. Toro, Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals, Math. Z., 299 (2021), 2131-2169.  doi: 10.1007/s00209-021-02719-5.

[9]

E. De Giorgi, Frontiere orientate di misura minima, Sem. Mat. Scuola Norm. Sup. Pisa, 1-56.

[10]

E. De Giorgi, Una estensione del teorema di Bernstein, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 3 (1965), 79-85. 

[11]

C. De Lellis, A. De Rosa and F. Ghiraldin, A direct approach to the anisotropic Plateau problem, Advances in Calculus of Variations, 12 (2019), 211-223, URL https://www.degruyter.com/view/journals/acv/12/2/article-p211.xml. doi: 10.1515/acv-2016-0057.

[12]

C. De LellisF. Ghiraldin and F. Maggi, A direct approach to Plateau's problem, Journal of the European Mathematical Society, 19 (2017), 2219-2240.  doi: 10.4171/JEMS/716.

[13]

G. De PhilippisA. De Rosa and F. Ghiraldin, A direct approach to Plateau's problem in any codimension, Advances in Mathematics, 288 (2016), 59-80.  doi: 10.1016/j.aim.2015.10.007.

[14]

G. De PhilippisA. De Rosa and F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148.  doi: 10.1002/cpa.21713.

[15]

G. De PhilippisA. De Rosa and F. Ghiraldin, Existence results for minimizers of parametric elliptic functionals, Journal of Geometric Analysis, 30 (2020), 1450-1465.  doi: 10.1007/s12220-019-00165-8.

[16]

O. S. de Queiroz and L. S. Tavares, Almost minimizers for semilinear free boundary problems with variable coefficients, Mathematische Nachrichten, 291 (2018), 1486-1501.  doi: 10.1002/mana.201600103.

[17]

A. De Rosa, Minimization of anisotropic energies in classes of rectifiable varifolds, SIAM Journal on Mathematical Analysis, 50 (2018), 162-181.  doi: 10.1137/17M1112479.

[18]

A. De Rosa and S. Kolasiński, Equivalence of the ellipticity conditions for geometric variational problems, Communications on Pure and Applied Mathematics, 73 (2020), 2473-2515.  doi: 10.1002/cpa.21890.

[19]

A. De Rosa and R. Tione, Regularity for graphs with bounded anisotropic mean curvature.,

[20]

J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc., 33 (1931), 263-321.  doi: 10.1090/S0002-9947-1931-1501590-9.

[21]

H. Federer, Geometric Measure Theory, vol. 1996, Classics in Mathematics, Springer New York, 1969. doi: 10.1007/978-3-642-62010-2.

[22]

H. Federer, 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.  doi: 10.1090/S0002-9904-1970-12542-3.

[23]

W. H. Fleming, On the oriented plateau problem, Rendiconti del Circolo Matematico di Palermo, 11 (1962), 69-90.  doi: 10.1007/BF02849427.

[24]

E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana University Mathematics Journal, 32 (1983), 25-37, URL http://www.jstor.org/stable/24893183. doi: 10.1512/iumj.1983.32.32003.

[25]

J. Harrison and H. Pugh, Existence and soap film regularity of solutions to Plateau's problem, Advances in Calculus of Variations, 9 (2016), 357-394.  doi: 10.1515/acv-2015-0023.

[26]

S. Jeon, A. Petrosyan and M. Smit Vega Garcia, Almost minimizers for the thin obstacle problem with variable coefficients, 2020.

[27]

D. King, F. Maggi and S. Stuvard, Plateau's problem as a singular limit of capillarity problems, 2020. doi: 10.1002/cpa.21951.

[28]

J. Lamboley and P. Sicbaldi, Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold, Journal de Mathématiques Pures et Appliqués, 141 (2020), 137-183.  doi: 10.1016/j.matpur.2020.06.006.

[29] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139108133.
[30] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, United Kingdom, 1995.  doi: 10.1017/CBO9780511623813.
[31]

M. Miranda, Sul minimo dell'integrale del gradiente di una fonzione, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1965), 626-665. 

[32]

H. R. Parks, Regularity of solutions to elliptic isoperimetric problems, Pacific J. Math., 113 (1984), 463-470, URL https://projecteuclid.org:443/euclid.pjm/1102709205. doi: 10.2140/pjm.1984.113.463.

[33]

T. Radó, On Plateau's problem, Annals of Mathematics, Second Series, 31 (1930), 457-469.  doi: 10.2307/1968237.

[34]

E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92.  doi: 10.1007/BF02547186.

[35]

E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Annals of Mathematics, Second Series, 80 (1964), 1-14.  doi: 10.2307/1970488.

[36]

E. R. Reifenberg, On the analyticity of minimal surfaces, Annals of Mathematics, Second Series, 80 (1964), 15-21.  doi: 10.2307/1970489.

[37]

R. Schoen and L. Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana University Mathematics Journal, 31 (1982), 415-434.  doi: 10.1512/iumj.1982.31.31035.

[38]

R. SchoenL. Simon and F. J. Almgren, Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals, Acta Mathematica, 139 (1977), 217-265.  doi: 10.1007/BF02392238.

[39]

J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.  doi: 10.2307/1970556.

[40]

D. Spector, Simple proofs of some results of Reshetnyak, Proc. Amer. Math. Soc., 139 (2011), 1681-1690.  doi: 10.1090/S0002-9939-2010-10593-2.

[41]

L. SpolaorB. Trey and B. Velichkov, Free boundary regularity for a multiphase shape optimization problem, Comm. Partial Differential Equations, 45 (2020), 77-108.  doi: 10.1080/03605302.2019.1658773.

[42]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector., Journal Für die Reine und Angewandte Mathematik, 334 (1982), 27-39, URL http://eudml.org/doc/152447. doi: 10.1515/crll.1982.334.27.

[43]

I. Tamanini, Regularity Results for Almost Minimal Oriented Hypersurfaces in $R^n$, Quaderni del Dipartimento Matematica dell'Universita de Lecce, 1984, URL https://books.google.com/books?id=YorovgEACAAJ.

[44]

J. E. Taylor, The structure of singularities in soap-bubble-Like and soap-film-like minimal surfaces, Annals of Mathematics, Second Series, 103 (1976), 489-539.  doi: 10.2307/1970949.

[45]

J. E. Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in $\mathbb{R}^3$, Annals of Mathematics, Second Series, 103 (1976), 541-546.  doi: 10.2307/1970950.

[46]

G. Wulff, Zur Frage der Geschwindigkeit des Wachsturms und der auflösungder Kristall-Flächen, Zeitschrift für Kristallographie, 34 (1901), 449-530.  doi: 10.1524/zkri.1901.34.1.449.

show all references

References:
[1]

W. K. Allard, On the first variation of a varifold, Ann. of Math., 95 (1972), 417-491.  doi: 10.2307/1970868.

[2]

W. K. Allard, A characterization of the area integrand, Symposia Mathematica, 14 (1974), 429-444. 

[3]

F. J. Almgren, Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem, Ann. of Math. (2), 84 (1966), 277-292.  doi: 10.2307/1970520.

[4]

F. J. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Annals of Mathematics, 87 (1968), 321-391.  doi: 10.2307/1970587.

[5]

E. Bombieri, Regularity theory for almost minimal currents, Arch. Rational Mech. Anal., 78 (1982), 99-130.  doi: 10.1007/BF00250836.

[6]

E. BombieriE. De Giorgi and E. Giusti, Minimal cones and the Bernstein problem, Ivent. Math., 7 (1969), 243-268.  doi: 10.1007/BF01404309.

[7]

S. Bortz, M. Engelstein, M. Goering, T. Toro and Z. Zhao, Two phase free boundary problem for Poisson kernels, 2019.

[8]

G. DavidM. EngelsteinM. Smit Vega Garcia and T. Toro, Regularity for almost-minimizers of variable coefficient Bernoulli-type functionals, Math. Z., 299 (2021), 2131-2169.  doi: 10.1007/s00209-021-02719-5.

[9]

E. De Giorgi, Frontiere orientate di misura minima, Sem. Mat. Scuola Norm. Sup. Pisa, 1-56.

[10]

E. De Giorgi, Una estensione del teorema di Bernstein, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 3 (1965), 79-85. 

[11]

C. De Lellis, A. De Rosa and F. Ghiraldin, A direct approach to the anisotropic Plateau problem, Advances in Calculus of Variations, 12 (2019), 211-223, URL https://www.degruyter.com/view/journals/acv/12/2/article-p211.xml. doi: 10.1515/acv-2016-0057.

[12]

C. De LellisF. Ghiraldin and F. Maggi, A direct approach to Plateau's problem, Journal of the European Mathematical Society, 19 (2017), 2219-2240.  doi: 10.4171/JEMS/716.

[13]

G. De PhilippisA. De Rosa and F. Ghiraldin, A direct approach to Plateau's problem in any codimension, Advances in Mathematics, 288 (2016), 59-80.  doi: 10.1016/j.aim.2015.10.007.

[14]

G. De PhilippisA. De Rosa and F. Ghiraldin, Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies, Communications on Pure and Applied Mathematics, 71 (2018), 1123-1148.  doi: 10.1002/cpa.21713.

[15]

G. De PhilippisA. De Rosa and F. Ghiraldin, Existence results for minimizers of parametric elliptic functionals, Journal of Geometric Analysis, 30 (2020), 1450-1465.  doi: 10.1007/s12220-019-00165-8.

[16]

O. S. de Queiroz and L. S. Tavares, Almost minimizers for semilinear free boundary problems with variable coefficients, Mathematische Nachrichten, 291 (2018), 1486-1501.  doi: 10.1002/mana.201600103.

[17]

A. De Rosa, Minimization of anisotropic energies in classes of rectifiable varifolds, SIAM Journal on Mathematical Analysis, 50 (2018), 162-181.  doi: 10.1137/17M1112479.

[18]

A. De Rosa and S. Kolasiński, Equivalence of the ellipticity conditions for geometric variational problems, Communications on Pure and Applied Mathematics, 73 (2020), 2473-2515.  doi: 10.1002/cpa.21890.

[19]

A. De Rosa and R. Tione, Regularity for graphs with bounded anisotropic mean curvature.,

[20]

J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc., 33 (1931), 263-321.  doi: 10.1090/S0002-9947-1931-1501590-9.

[21]

H. Federer, Geometric Measure Theory, vol. 1996, Classics in Mathematics, Springer New York, 1969. doi: 10.1007/978-3-642-62010-2.

[22]

H. Federer, 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.  doi: 10.1090/S0002-9904-1970-12542-3.

[23]

W. H. Fleming, On the oriented plateau problem, Rendiconti del Circolo Matematico di Palermo, 11 (1962), 69-90.  doi: 10.1007/BF02849427.

[24]

E. Gonzalez, U. Massari and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana University Mathematics Journal, 32 (1983), 25-37, URL http://www.jstor.org/stable/24893183. doi: 10.1512/iumj.1983.32.32003.

[25]

J. Harrison and H. Pugh, Existence and soap film regularity of solutions to Plateau's problem, Advances in Calculus of Variations, 9 (2016), 357-394.  doi: 10.1515/acv-2015-0023.

[26]

S. Jeon, A. Petrosyan and M. Smit Vega Garcia, Almost minimizers for the thin obstacle problem with variable coefficients, 2020.

[27]

D. King, F. Maggi and S. Stuvard, Plateau's problem as a singular limit of capillarity problems, 2020. doi: 10.1002/cpa.21951.

[28]

J. Lamboley and P. Sicbaldi, Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold, Journal de Mathématiques Pures et Appliqués, 141 (2020), 137-183.  doi: 10.1016/j.matpur.2020.06.006.

[29] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139108133.
[30] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, vol. 44 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, United Kingdom, 1995.  doi: 10.1017/CBO9780511623813.
[31]

M. Miranda, Sul minimo dell'integrale del gradiente di una fonzione, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 19 (1965), 626-665. 

[32]

H. R. Parks, Regularity of solutions to elliptic isoperimetric problems, Pacific J. Math., 113 (1984), 463-470, URL https://projecteuclid.org:443/euclid.pjm/1102709205. doi: 10.2140/pjm.1984.113.463.

[33]

T. Radó, On Plateau's problem, Annals of Mathematics, Second Series, 31 (1930), 457-469.  doi: 10.2307/1968237.

[34]

E. R. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta Math., 104 (1960), 1-92.  doi: 10.1007/BF02547186.

[35]

E. R. Reifenberg, An epiperimetric inequality related to the analyticity of minimal surfaces, Annals of Mathematics, Second Series, 80 (1964), 1-14.  doi: 10.2307/1970488.

[36]

E. R. Reifenberg, On the analyticity of minimal surfaces, Annals of Mathematics, Second Series, 80 (1964), 15-21.  doi: 10.2307/1970489.

[37]

R. Schoen and L. Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana University Mathematics Journal, 31 (1982), 415-434.  doi: 10.1512/iumj.1982.31.31035.

[38]

R. SchoenL. Simon and F. J. Almgren, Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals, Acta Mathematica, 139 (1977), 217-265.  doi: 10.1007/BF02392238.

[39]

J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math., 88 (1968), 62-105.  doi: 10.2307/1970556.

[40]

D. Spector, Simple proofs of some results of Reshetnyak, Proc. Amer. Math. Soc., 139 (2011), 1681-1690.  doi: 10.1090/S0002-9939-2010-10593-2.

[41]

L. SpolaorB. Trey and B. Velichkov, Free boundary regularity for a multiphase shape optimization problem, Comm. Partial Differential Equations, 45 (2020), 77-108.  doi: 10.1080/03605302.2019.1658773.

[42]

I. Tamanini, Boundaries of Caccioppoli sets with Hölder-continuous normal vector., Journal Für die Reine und Angewandte Mathematik, 334 (1982), 27-39, URL http://eudml.org/doc/152447. doi: 10.1515/crll.1982.334.27.

[43]

I. Tamanini, Regularity Results for Almost Minimal Oriented Hypersurfaces in $R^n$, Quaderni del Dipartimento Matematica dell'Universita de Lecce, 1984, URL https://books.google.com/books?id=YorovgEACAAJ.

[44]

J. E. Taylor, The structure of singularities in soap-bubble-Like and soap-film-like minimal surfaces, Annals of Mathematics, Second Series, 103 (1976), 489-539.  doi: 10.2307/1970949.

[45]

J. E. Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in $\mathbb{R}^3$, Annals of Mathematics, Second Series, 103 (1976), 541-546.  doi: 10.2307/1970950.

[46]

G. Wulff, Zur Frage der Geschwindigkeit des Wachsturms und der auflösungder Kristall-Flächen, Zeitschrift für Kristallographie, 34 (1901), 449-530.  doi: 10.1524/zkri.1901.34.1.449.

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