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December  2019, 39(12): 6877-6912. doi: 10.3934/dcds.2019236

Soap films with gravity and almost-minimal surfaces

Francesco Maggi 1, , Salvatore Stuvard 1, and  Antonello Scardicchio 2,

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

Received  July 2018 Revised  January 2019 Published  June 2019

Fund Project: F. M. and S. S. have been supported by NSF Grants DMS-1565354, DMS-1361122 and DMS-1262411.

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.

Keywords: Minimal surfaces, Plateau problem, capillarity theory, integral currents, integral varifolds.
Mathematics Subject Classification: Primary: 49Q05, 49Q15; Secondary: 53A10.
Citation: Francesco Maggi, Salvatore Stuvard, Antonello Scardicchio. Soap films with gravity and almost-minimal surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 6877-6912. doi: 10.3934/dcds.2019236
References:
[1]

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

[2]

W. K. Allard, On the first variation of a varifold: boundary behaviour, Ann. Math., 101 (1975), 418-446.  doi: 10.2307/1970934.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2), 72 (1960), 458-520.  doi: 10.2307/1970227.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

T. Young, An essay on the cohesion of fluids, Philos. Trans. Roy. Soc. London, 65–87. doi: 10.1098/rspl.1800.0095.  Google Scholar

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.  Google Scholar

[2]

W. K. Allard, On the first variation of a varifold: boundary behaviour, Ann. Math., 101 (1975), 418-446.  doi: 10.2307/1970934.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[22]

H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math. (2), 72 (1960), 458-520.  doi: 10.2307/1970227.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795557.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

T. Young, An essay on the cohesion of fluids, Philos. Trans. Roy. Soc. London, 65–87. doi: 10.1098/rspl.1800.0095.  Google Scholar

Figure 1.  On the left, a boundary $ \Gamma $, consisting of three circles, that is accessible from infinity. The acute wedges realizing the inclusions 3 are depicted by dashed lines. Notice that it is not necessary that $ \Gamma $ is contained into a convex set, or into a mean convex set, for the condition to hold. On the right, another set of circles defining a boundary $ \Gamma $ which does not satisfy accessibility from infinity. Indeed, there is no way to touch the smaller circle with an acute wedge containing the larger ones
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Figure 5.  The construction described in Example 7
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Figure 4.  When Γ consists of two parallel disks there are, in addition to the disconnected surface defined by two disks, four minimal surfaces, two of them singular, all composed by joining pieces of catenoids
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Figure 3.  Using Gauss' capillarity energy to formulate Plateau's problem. Minimization of $ \sigma\,\mathcal{H}^2(M) $ among surfaces with $ \partial M = \Gamma $ is replaced by minimizing the capillarity energy among regions contained in the complement of a $ \delta $-neighborhood of $ \Gamma $. Equilibrium configurations with volume $ \varepsilon\ll\delta\,\mathcal{H}^2(S)\ll1 $ arise as normal neighborhoods of minimal surfaces spanned by $ \Gamma $. Here $ S $ denotes the boundary of $ E $ away from the wire frame
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Figure 2.  The derivation of 12, after [16, Section Ⅰ.4]
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Figure 6.  Bubbling is possible even when $\Gamma$ is accessible from infinity if a weak notion of deficit is used. Here $M_j$ is the surface of revolution obtained by rotating the one-dimensional profile on the right, $B_{\varepsilon _j}(\Gamma_1)$ denotes an $\varepsilon _j$-neighborhood of the circle $\Gamma_1$, and $M_j^*$ is the part of $M_j$ lying outside $B_{\varepsilon _j}(\Gamma_1)$. We take $\varepsilon _j$ such that $M_j$ intersects $\partial B_{\varepsilon _j}(\Gamma_1)$ in three circles, and so that the $H_{M_j}$ is uniformly small on $M_j\setminus M_j^*$. The limit surface counts one copy of $K$, and two copies of the disk filling $\Gamma_1$
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