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Regularity results for the equation $ u_{11}u_{22} = 1 $
Soap films with gravity and almost-minimal surfaces
1. | Department of Mathematics, The University of Texas at Austin, 2515 Speedway STOP C1200, Austin, TX 78712, USA |
2. | International Centre for Theoretical Physics, Strada Costiera 11, Trieste 34151, Italy |
Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.
References:
[1] |
W. K. Allard,
On the first variation of a varifold, Ann. Math., 95 (1972), 417-491.
doi: 10.2307/1970868. |
[2] |
W. K. Allard,
On the first variation of a varifold: boundary behaviour, Ann. Math., 101 (1975), 418-446.
doi: 10.2307/1970934. |
[3] |
S. Amato, G. Bellettini and M. Paolini,
Constrained BV functions on covering spaces for minimal networks and Plateau's type problems, Adv. Calc. Var., 10 (2017), 25-47.
doi: 10.1515/acv-2015-0021. |
[4] |
H. Brezis and J.-M. Coron,
Multiple solutions of $H$-systems and Rellich's conjecture, Comm. Pure Appl. Math., 37 (1984), 149-187.
doi: 10.1002/cpa.3160370202. |
[5] |
M. Cicalese, G. P. Leonardi and F. Maggi,
Improved convergence theorems for bubble clusters I. The planar case, Indiana Univ. Math. J., 65 (2016), 1979-2050.
doi: 10.1512/iumj.2016.65.5932. |
[6] |
G. Ciraolo and F. Maggi,
On the shape of compact hypersurfaces with almost-constant mean curvature, Comm. Pure Appl. Math., 70 (2017), 665-716.
doi: 10.1002/cpa.21683. |
[7] |
C. Cohen, B. Darbois Texier, E. Reyssat, J. H. Snoeijer, D. Quéré and C. Clanet,
On the shape of giant soap bubbles, Proceedings of the National Academy of Sciences, 114 (2017), 2515-2519.
doi: 10.1073/pnas.1616904114. |
[8] |
G. David, Should we solve Plateau's problem again?, in Advances in analysis: The legacy of Elias M. Stein, Princeton Math. Ser., 50, Princeton Univ. Press, Princeton, NJ, 2014,108-145. |
[9] |
P.-G. de Gennes, F. Brochard-Wyart and D. Quéré, Capillarity and Wetting Phenomena, Translated by A. Reisinger, Springer, 2003.
doi: 10.1007/978-0-387-21656-0. |
[10] |
C. De Lellis, A. De Rosa and F. Ghiraldin, A direct approach to the anisotropic plateau problem, Adv. Calc. Var.
doi: 10.1515/acv-2016-0057. |
[11] |
C. De Lellis, F. Ghiraldin and F. Maggi,
A direct approach to Plateau's problem, J. Eur. Math. Soc. (JEMS), 19 (2017), 2219-2240.
doi: 10.4171/JEMS/716. |
[12] |
C. De Lellis and J. Ramic, Min-max theory for minimal hypersurfaces with boundary, preprint, arXiv: 1611.00926, to appear in Jour. Ann. Inst. Fourier. Google Scholar |
[13] |
G. De Philippis, A. De Rosa and F. Ghiraldin,
A direct approach to Plateau's problem in any codimension, Adv. Math., 288 (2016), 59-80.
doi: 10.1016/j.aim.2015.10.007. |
[14] |
G. De Philippis and F. Maggi,
Sharp stability inequalities for the Plateau problem, J. Differential Geom., 96 (2014), 399-456.
doi: 10.4310/jdg/1395321846. |
[15] |
A. De Rosa,
Minimization of anisotropic energies in classes of rectifiable varifolds, SIAM J. Math. Anal., 50 (2018), 162-181.
doi: 10.1137/17M1112479. |
[16] |
R. Defay and I. Prigogine, Surface Tension and Adsorption, Translated by D. G. Everett, John Wiley and sons, Inc., New York, NY, 1966. Google Scholar |
[17] |
M. G. Delgadino and F. Maggi, Alexandrov's theorem revisited, preprint, arXiv: 1711.07690.
doi: 10.2140/apde.2019.12.1613. |
[18] |
M. G. Delgadino, F. Maggi, C. Mihaila and R. Neumayer,
Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals, Arch. Ration. Mech. Anal., 230 (2018), 1131-1177.
doi: 10.1007/s00205-018-1267-8. |
[19] |
F. Duzaar and M. Fuchs,
On the existence of integral currents with prescribed mean curvature vector, Manuscripta Math., 67 (1990), 41-67.
doi: 10.1007/BF02568422. |
[20] |
F. Duzaar and M. Fuchs,
A general existence theorem for integral currents with prescribed mean curvature form, Boll. Un. Mat. Ital. B (7), 6 (1992), 901-912.
|
[21] |
Y. Fang and S. Kolasinski, Existence of solutions to a general geometric elliptic variational problem, Calc. Var. Partial Differential Equations, 57 (2018), 91.
doi: 10.1007/s00526-018-1348-4. |
[22] |
H. Federer and W. H. Fleming,
Normal and integral currents, Ann. of Math. (2), 72 (1960), 458-520.
doi: 10.2307/1970227. |
[23] |
A. Figalli and F. Maggi,
On the shape of liquid drops and crystals in the small mass regime, Arch. Rat. Mech. Anal., 201 (2011), 143-207.
doi: 10.1007/s00205-010-0383-x. |
[24] |
C. F. Gauss, Principia generalia theoriae figurae fluidorum, Comment. Soc. Regiae Scient. Gottingensis Rec. Google Scholar |
[25] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin; New York, 1998. |
[26] |
G. G. Giusteri, L. Lussardi and E. Fried,
Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci., 27 (2017), 1043-1063.
doi: 10.1007/s00332-017-9359-4. |
[27] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/9789812795557. |
[28] |
J. Harrison,
On Plateau's problem for soap films with a bound on energy, J. Geom. Anal., 14 (2004), 319-329.
doi: 10.1007/BF02922075. |
[29] |
J. Harrison and H. Pugh,
Existence and soap film regularity of solutions to Plateau's problem, Adv. Calc. Var., 9 (2016), 357-394.
doi: 10.1515/acv-2015-0023. |
[30] |
G. Huisken,
Nonparametric mean curvature evolution with boundary conditions, J. Differential Equations, 77 (1989), 369-378.
doi: 10.1016/0022-0396(89)90149-6. |
[31] |
B. Krummel and F. Maggi, Isoperimetry with upper mean curvature bounds and sharp stability estimates, Calc. Var. Partial Differential Equations, 56 (2017), Art. 53, 43.
doi: 10.1007/s00526-017-1139-3. |
[32] |
P. S. Laplace, Mécanique céleste, 1806, Suppl. 10th volume. Google Scholar |
[33] |
G. P. Leonardi and F. Maggi,
Improved convergence theorems for bubble clusters II. The three-dimensional case, Indiana Univ. Math. J., 66 (2017), 559-608.
doi: 10.1512/iumj.2017.66.6016. |
[34] |
R. Schätzle,
Quadratic tilt-excess decay and strong maximum principle for varifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3 (2004), 171-231.
|
[35] |
L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. |
[36] |
J. Spruck,
Interior gradient estimates and existence theorems for constant mean curvature graphs in $M^n\times\bf R$, Pure Appl. Math. Q., 3 (2007), 785-800.
doi: 10.4310/PAMQ.2007.v3.n3.a6. |
[37] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[38] |
B. White,
Currents and flat chains associated to varifolds, with an application to mean curvature flow, Duke Math. J., 148 (2009), 41-62.
doi: 10.1215/00127094-2009-019. |
[39] |
T. Young, An essay on the cohesion of fluids, Philos. Trans. Roy. Soc. London, 65–87.
doi: 10.1098/rspl.1800.0095. |
show all references
References:
[1] |
W. K. Allard,
On the first variation of a varifold, Ann. Math., 95 (1972), 417-491.
doi: 10.2307/1970868. |
[2] |
W. K. Allard,
On the first variation of a varifold: boundary behaviour, Ann. Math., 101 (1975), 418-446.
doi: 10.2307/1970934. |
[3] |
S. Amato, G. Bellettini and M. Paolini,
Constrained BV functions on covering spaces for minimal networks and Plateau's type problems, Adv. Calc. Var., 10 (2017), 25-47.
doi: 10.1515/acv-2015-0021. |
[4] |
H. Brezis and J.-M. Coron,
Multiple solutions of $H$-systems and Rellich's conjecture, Comm. Pure Appl. Math., 37 (1984), 149-187.
doi: 10.1002/cpa.3160370202. |
[5] |
M. Cicalese, G. P. Leonardi and F. Maggi,
Improved convergence theorems for bubble clusters I. The planar case, Indiana Univ. Math. J., 65 (2016), 1979-2050.
doi: 10.1512/iumj.2016.65.5932. |
[6] |
G. Ciraolo and F. Maggi,
On the shape of compact hypersurfaces with almost-constant mean curvature, Comm. Pure Appl. Math., 70 (2017), 665-716.
doi: 10.1002/cpa.21683. |
[7] |
C. Cohen, B. Darbois Texier, E. Reyssat, J. H. Snoeijer, D. Quéré and C. Clanet,
On the shape of giant soap bubbles, Proceedings of the National Academy of Sciences, 114 (2017), 2515-2519.
doi: 10.1073/pnas.1616904114. |
[8] |
G. David, Should we solve Plateau's problem again?, in Advances in analysis: The legacy of Elias M. Stein, Princeton Math. Ser., 50, Princeton Univ. Press, Princeton, NJ, 2014,108-145. |
[9] |
P.-G. de Gennes, F. Brochard-Wyart and D. Quéré, Capillarity and Wetting Phenomena, Translated by A. Reisinger, Springer, 2003.
doi: 10.1007/978-0-387-21656-0. |
[10] |
C. De Lellis, A. De Rosa and F. Ghiraldin, A direct approach to the anisotropic plateau problem, Adv. Calc. Var.
doi: 10.1515/acv-2016-0057. |
[11] |
C. De Lellis, F. Ghiraldin and F. Maggi,
A direct approach to Plateau's problem, J. Eur. Math. Soc. (JEMS), 19 (2017), 2219-2240.
doi: 10.4171/JEMS/716. |
[12] |
C. De Lellis and J. Ramic, Min-max theory for minimal hypersurfaces with boundary, preprint, arXiv: 1611.00926, to appear in Jour. Ann. Inst. Fourier. Google Scholar |
[13] |
G. De Philippis, A. De Rosa and F. Ghiraldin,
A direct approach to Plateau's problem in any codimension, Adv. Math., 288 (2016), 59-80.
doi: 10.1016/j.aim.2015.10.007. |
[14] |
G. De Philippis and F. Maggi,
Sharp stability inequalities for the Plateau problem, J. Differential Geom., 96 (2014), 399-456.
doi: 10.4310/jdg/1395321846. |
[15] |
A. De Rosa,
Minimization of anisotropic energies in classes of rectifiable varifolds, SIAM J. Math. Anal., 50 (2018), 162-181.
doi: 10.1137/17M1112479. |
[16] |
R. Defay and I. Prigogine, Surface Tension and Adsorption, Translated by D. G. Everett, John Wiley and sons, Inc., New York, NY, 1966. Google Scholar |
[17] |
M. G. Delgadino and F. Maggi, Alexandrov's theorem revisited, preprint, arXiv: 1711.07690.
doi: 10.2140/apde.2019.12.1613. |
[18] |
M. G. Delgadino, F. Maggi, C. Mihaila and R. Neumayer,
Bubbling with $L^2$-almost constant mean curvature and an Alexandrov-type theorem for crystals, Arch. Ration. Mech. Anal., 230 (2018), 1131-1177.
doi: 10.1007/s00205-018-1267-8. |
[19] |
F. Duzaar and M. Fuchs,
On the existence of integral currents with prescribed mean curvature vector, Manuscripta Math., 67 (1990), 41-67.
doi: 10.1007/BF02568422. |
[20] |
F. Duzaar and M. Fuchs,
A general existence theorem for integral currents with prescribed mean curvature form, Boll. Un. Mat. Ital. B (7), 6 (1992), 901-912.
|
[21] |
Y. Fang and S. Kolasinski, Existence of solutions to a general geometric elliptic variational problem, Calc. Var. Partial Differential Equations, 57 (2018), 91.
doi: 10.1007/s00526-018-1348-4. |
[22] |
H. Federer and W. H. Fleming,
Normal and integral currents, Ann. of Math. (2), 72 (1960), 458-520.
doi: 10.2307/1970227. |
[23] |
A. Figalli and F. Maggi,
On the shape of liquid drops and crystals in the small mass regime, Arch. Rat. Mech. Anal., 201 (2011), 143-207.
doi: 10.1007/s00205-010-0383-x. |
[24] |
C. F. Gauss, Principia generalia theoriae figurae fluidorum, Comment. Soc. Regiae Scient. Gottingensis Rec. Google Scholar |
[25] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin; New York, 1998. |
[26] |
G. G. Giusteri, L. Lussardi and E. Fried,
Solution of the Kirchhoff-Plateau problem, J. Nonlinear Sci., 27 (2017), 1043-1063.
doi: 10.1007/s00332-017-9359-4. |
[27] |
E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
doi: 10.1142/9789812795557. |
[28] |
J. Harrison,
On Plateau's problem for soap films with a bound on energy, J. Geom. Anal., 14 (2004), 319-329.
doi: 10.1007/BF02922075. |
[29] |
J. Harrison and H. Pugh,
Existence and soap film regularity of solutions to Plateau's problem, Adv. Calc. Var., 9 (2016), 357-394.
doi: 10.1515/acv-2015-0023. |
[30] |
G. Huisken,
Nonparametric mean curvature evolution with boundary conditions, J. Differential Equations, 77 (1989), 369-378.
doi: 10.1016/0022-0396(89)90149-6. |
[31] |
B. Krummel and F. Maggi, Isoperimetry with upper mean curvature bounds and sharp stability estimates, Calc. Var. Partial Differential Equations, 56 (2017), Art. 53, 43.
doi: 10.1007/s00526-017-1139-3. |
[32] |
P. S. Laplace, Mécanique céleste, 1806, Suppl. 10th volume. Google Scholar |
[33] |
G. P. Leonardi and F. Maggi,
Improved convergence theorems for bubble clusters II. The three-dimensional case, Indiana Univ. Math. J., 66 (2017), 559-608.
doi: 10.1512/iumj.2017.66.6016. |
[34] |
R. Schätzle,
Quadratic tilt-excess decay and strong maximum principle for varifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 3 (2004), 171-231.
|
[35] |
L. Simon, Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. |
[36] |
J. Spruck,
Interior gradient estimates and existence theorems for constant mean curvature graphs in $M^n\times\bf R$, Pure Appl. Math. Q., 3 (2007), 785-800.
doi: 10.4310/PAMQ.2007.v3.n3.a6. |
[37] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[38] |
B. White,
Currents and flat chains associated to varifolds, with an application to mean curvature flow, Duke Math. J., 148 (2009), 41-62.
doi: 10.1215/00127094-2009-019. |
[39] |
T. Young, An essay on the cohesion of fluids, Philos. Trans. Roy. Soc. London, 65–87.
doi: 10.1098/rspl.1800.0095. |





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