• Previous Article
    Systems of semilinear wave equations with multiple speeds in two space dimensions and a weaker null condition
  • CPAA Home
  • This Issue
  • Next Article
    Strongly singular convective elliptic equations in $ \mathbb{R}^N $ driven by a non-homogeneous operator
doi: 10.3934/cpaa.2022086
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Plateau-rayleigh instability of singular minimal surfaces

Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain

Received  August 2021 Revised  April 2022 Early access May 2022

Fund Project: This work has been partially supported by the Projects I+D+i PID2020-117868GB-I00, A-FQM-139-UGR18 and P18-FR-4049

We prove a Plateau-Rayleigh criterion of instability for singular minimal surfaces, providing explicit bounds on the amplitude and length of the surface. More generally, we study the stability of $ \alpha $-singular minimal hypersurfaces considered as hypersurfaces in weighted manifolds. If $ \alpha<0 $ and the hypersurface is a graph, then we prove that the hypersurface is stable. If $ \alpha>0 $ and the surface is cylindrical, we give numerical evidences of the instability of long cylindrical $ \alpha $-singular minimal surfaces.

Citation: Rafael López. Plateau-rayleigh instability of singular minimal surfaces. Communications on Pure and Applied Analysis, doi: 10.3934/cpaa.2022086
References:
[1]

V. Bayle, Propriétés de Concavité du Profil Isopérimétrique et Applications, Ph.D. Thesis, Institut Joseph Fourier, Grenoble, 2004.

[2]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-74311-8.

[3]

R. BöhmeS. Hildebrandt and E. Taush, The two-dimensional analogue of the catenary, Pacific J. Math., 88 (1980), 247-278. 

[4]

K. Castro and C. Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, J. Geom. Phys., 79 (2014), 14-28.  doi: 10.1016/j.geomphys.2014.01.013.

[5]

T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, Ann. Math., 175 (2012), 755-833.  doi: 10.4007/annals.2012.175.2.7.

[6]

L. Colter, Cylindrical liquid bridges, Involve, 8 (2015), 695-705.  doi: 10.2140/involve.2015.8.695.

[7]

U. Dierkes, A Bernstein result for energy minimizing hypersurfaces, Calc. Var. Partial Differ. Equ., 1 (1993), 37-54.  doi: 10.1007/BF02163263.

[8]

U. Dierkes, Singular minimal surfaces, in Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003.

[9]

U. Dierkes and G. Huisken, The $n$-dimensional analogue of the catenary: existence and nonexistence, Pacific J. Math., 141 (1990), 47-54. 

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998.

[11]

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Commun. Pure Appl. Math., 33 (1980), 199-211.  doi: 10.1002/cpa.3160330206.

[12]

M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178-215.  doi: 10.1007/s000390300004.

[13]

R. López, A criterion on instability of cylindrical rotating surfaces, Archiv Math., 94 (2010), 91-99.  doi: 10.1007/s00013-009-0085-5.

[14]

R. López, Bifurcation of cylinders for wetting and dewetting models with striped geometry, SIAM J. Math. Anal., 44 (2012), 946-965.  doi: 10.1137/11082484X.

[15]

R. López, Invariant singular minimal surfaces, Ann. Global Anal. Geom., 53 (2018), 521-541.  doi: 10.1007/s10455-017-9586-9.

[16]

J. McCuan, Extremities of stability for pendant drops, in Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/conm/599/11944.

[17]

J. McCuan, The stability of cylindrical pendant drops, Mem. Amer. Math. Soc., 250 (2017), no. 1189. doi: 10.1090/memo/1189.

[18]

F. Otto, Zugbeanspruchte Konstruktionen, Berlin, Frankfurt, Wien: Ullstein, 1962.

[19]

B. Palmer and O. Perdomo, Equilibrium shapes of cylindrical rotating liquid drops, Bull. Braz. Math. Soc., 46 (2015), 515-561. 

[20]

J. A. F. Plateau, Statique Expérimentale et Théorique Des Liquides Soumis Aux Seules Forces Moléculaires, vol. 2. Gauthier-Villars, 2018.

[21]

J. W. S. Rayleigh, On the instability of jets, Proc. London Math. Soc., 10 (1879), 4-13.  doi: 10.1112/plms/s1-10.1.4.

[22]

R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, in Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, 1983.

[23]

L. Shahriyari, Translating graphs by mean curvature flow, Geom Dedicata, 175 (2015), 57-64.  doi: 10.1007/s10711-014-0028-6.

[24]

J. Sun, Lagrangian L-stability of Lagrangian translating solitons, Manuscripta Math., 161 (2020), 247-255.  doi: 10.1007/s00229-018-1089-x.

[25]

Wikipedia, Colegio Teresiano de Barcelona, "https://es.wikipedia.org/w/index.php?title=Colegio_Teresiano_de_Barcelona&oldid=134544852".

show all references

References:
[1]

V. Bayle, Propriétés de Concavité du Profil Isopérimétrique et Applications, Ph.D. Thesis, Institut Joseph Fourier, Grenoble, 2004.

[2]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-540-74311-8.

[3]

R. BöhmeS. Hildebrandt and E. Taush, The two-dimensional analogue of the catenary, Pacific J. Math., 88 (1980), 247-278. 

[4]

K. Castro and C. Rosales, Free boundary stable hypersurfaces in manifolds with density and rigidity results, J. Geom. Phys., 79 (2014), 14-28.  doi: 10.1016/j.geomphys.2014.01.013.

[5]

T. H. Colding and W. P. Minicozzi II, Generic mean curvature flow I; generic singularities, Ann. Math., 175 (2012), 755-833.  doi: 10.4007/annals.2012.175.2.7.

[6]

L. Colter, Cylindrical liquid bridges, Involve, 8 (2015), 695-705.  doi: 10.2140/involve.2015.8.695.

[7]

U. Dierkes, A Bernstein result for energy minimizing hypersurfaces, Calc. Var. Partial Differ. Equ., 1 (1993), 37-54.  doi: 10.1007/BF02163263.

[8]

U. Dierkes, Singular minimal surfaces, in Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 2003.

[9]

U. Dierkes and G. Huisken, The $n$-dimensional analogue of the catenary: existence and nonexistence, Pacific J. Math., 141 (1990), 47-54. 

[10]

L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, RI, 1998.

[11]

D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvature, Commun. Pure Appl. Math., 33 (1980), 199-211.  doi: 10.1002/cpa.3160330206.

[12]

M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13 (2003), 178-215.  doi: 10.1007/s000390300004.

[13]

R. López, A criterion on instability of cylindrical rotating surfaces, Archiv Math., 94 (2010), 91-99.  doi: 10.1007/s00013-009-0085-5.

[14]

R. López, Bifurcation of cylinders for wetting and dewetting models with striped geometry, SIAM J. Math. Anal., 44 (2012), 946-965.  doi: 10.1137/11082484X.

[15]

R. López, Invariant singular minimal surfaces, Ann. Global Anal. Geom., 53 (2018), 521-541.  doi: 10.1007/s10455-017-9586-9.

[16]

J. McCuan, Extremities of stability for pendant drops, in Geometric Analysis, Mathematical Relativity, and Nonlinear Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2013. doi: 10.1090/conm/599/11944.

[17]

J. McCuan, The stability of cylindrical pendant drops, Mem. Amer. Math. Soc., 250 (2017), no. 1189. doi: 10.1090/memo/1189.

[18]

F. Otto, Zugbeanspruchte Konstruktionen, Berlin, Frankfurt, Wien: Ullstein, 1962.

[19]

B. Palmer and O. Perdomo, Equilibrium shapes of cylindrical rotating liquid drops, Bull. Braz. Math. Soc., 46 (2015), 515-561. 

[20]

J. A. F. Plateau, Statique Expérimentale et Théorique Des Liquides Soumis Aux Seules Forces Moléculaires, vol. 2. Gauthier-Villars, 2018.

[21]

J. W. S. Rayleigh, On the instability of jets, Proc. London Math. Soc., 10 (1879), 4-13.  doi: 10.1112/plms/s1-10.1.4.

[22]

R. Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, in Seminar on Minimal Submanifolds, Princeton Univ. Press, Princeton, 1983.

[23]

L. Shahriyari, Translating graphs by mean curvature flow, Geom Dedicata, 175 (2015), 57-64.  doi: 10.1007/s10711-014-0028-6.

[24]

J. Sun, Lagrangian L-stability of Lagrangian translating solitons, Manuscripta Math., 161 (2020), 247-255.  doi: 10.1007/s00229-018-1089-x.

[25]

Wikipedia, Colegio Teresiano de Barcelona, "https://es.wikipedia.org/w/index.php?title=Colegio_Teresiano_de_Barcelona&oldid=134544852".

Figure 1.  Left: corridor in the Colegio de las Teresianas, Barcelona ([25]). Right: the singular minimal surface $ \{(s,t,\cosh(s)):s,t\in{\mathbb R}\} $ constructed by repeating a catenary (blue) in a horizontal direction
Figure 2.  Solutions of (2.3) for values $ \alpha = -2 $ (right), $ \alpha = 0.8 $ (middle) and $ \alpha = 2 $ (right)
Figure 3.  Case $ \alpha = 1 $. Left: the function $ I_1(a) $. Right: the function $ L\mapsto I_2(a,L) $ (here $ a = 1 $)
Figure 4.  The function $ L_0 = L_0(a) $ given in (4.6)
Table 1.  Values of $ I(a,L) $ for $ \alpha = 2 $
$ L $ $ 1 $ $ 10 $ $ 20 $ $ 30 $ $ 40 $ $ 50 $
$ a $
0.2 -0.02389 -0.02031 -0.02028 -0.02028 -0.02028 -0.02028
0.3 -0.09371 -0.06398 -0.06375 -0.06371 -0.06370 -0.06369
0.4 -0.27824 -0.13500 -0.13392 -0.13372 -0.13365 -0.13362
0.5 -0.74222 -0.21609 -0.21210 -0.21137 -0.21111 -0.21099
0.6 -1.93468 -0.253826 -0.24109 -0.23873 -0.23790 -0.237527
0.7 -5.20226 -0.12268 -0.08420 -0.07708 -0.07458 -0.07343
0.8 -15.15280 $\fbox{0.45027} $ 0.5684 0.5903 0.59802 0.60157
0.9 -50.9705 $ \fbox{2.0161}$ 2.41751 2.49185 2.51786 2.52991
$ L $ $ 1 $ $ 10 $ $ 20 $ $ 30 $ $ 40 $ $ 50 $
$ a $
0.2 -0.02389 -0.02031 -0.02028 -0.02028 -0.02028 -0.02028
0.3 -0.09371 -0.06398 -0.06375 -0.06371 -0.06370 -0.06369
0.4 -0.27824 -0.13500 -0.13392 -0.13372 -0.13365 -0.13362
0.5 -0.74222 -0.21609 -0.21210 -0.21137 -0.21111 -0.21099
0.6 -1.93468 -0.253826 -0.24109 -0.23873 -0.23790 -0.237527
0.7 -5.20226 -0.12268 -0.08420 -0.07708 -0.07458 -0.07343
0.8 -15.15280 $\fbox{0.45027} $ 0.5684 0.5903 0.59802 0.60157
0.9 -50.9705 $ \fbox{2.0161}$ 2.41751 2.49185 2.51786 2.52991
Table 2.  Case $ \alpha = 2 $. Values of $ I(0.72,L) $
$ L $ $ 22 $ $ 24 $ $ 26 $ $ 28 $ $ 30 $ $ 32 $
-0.0053 -0.0032 -0.0015 -0.0002 $ \fbox{0.0007}$ 0.0016
$ L $ $ 22 $ $ 24 $ $ 26 $ $ 28 $ $ 30 $ $ 32 $
-0.0053 -0.0032 -0.0015 -0.0002 $ \fbox{0.0007}$ 0.0016
Table 3.  Case $ \alpha = 2 $. Values of $ I(a,L) $ when $ a>0.72 $
$ L $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ 7 8 9
$ a $
0.75 -2.061 -0.820 -0.386 -0.185 -0.076 -0.010 $\fbox{0.032} $ 0.061
0.80 -3.332 -1.143 -0.377 -0.022 $\fbox{0.170} $ 0.286 0.361 0.413
0.90 -10.829 -3.395 -0.793 $ \fbox{0.410}$ 1.064 1.459 1.715 1.890
1.00 -49.743 -17.942 -6.812 -1.660 $\fbox{1.137} $ 2.824 3.920 4.670
1.05 -126.723 -49.655 -22.681 -10.196 -3.414 $ \fbox{0.674}$ 3.328 5.148
$ L $ $ 2 $ $ 3 $ $ 4 $ $ 5 $ $ 6 $ 7 8 9
$ a $
0.75 -2.061 -0.820 -0.386 -0.185 -0.076 -0.010 $\fbox{0.032} $ 0.061
0.80 -3.332 -1.143 -0.377 -0.022 $\fbox{0.170} $ 0.286 0.361 0.413
0.90 -10.829 -3.395 -0.793 $ \fbox{0.410}$ 1.064 1.459 1.715 1.890
1.00 -49.743 -17.942 -6.812 -1.660 $\fbox{1.137} $ 2.824 3.920 4.670
1.05 -126.723 -49.655 -22.681 -10.196 -3.414 $ \fbox{0.674}$ 3.328 5.148
Table 4.  Case $ \alpha = 3 $. Values of $ I(a,L) $ when $ a>0.53 $
$ L $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ 10 11 12
$ a $
0.54 -0.394 -0.216 -0.119 -0.060 -0.022 $ \fbox{0.003}$ 0.021 0.035 0.046
0.56 -0.531 -0.247 -0.093 $\fbox{0.000} $ 0.059 0.100 0.130 0.152 0.168
0.58 -0.906 -0.433 -0.176 -0.021 $\fbox{0.079} $ 0.148 0.197 0.234 0.262
0.60 -1.928 -1.079 -0.618 -0.340 -0.160 -0.036 $ \fbox{0.051}$ 0.117 0.167
$ L $ $ 4 $ $ 5 $ $ 6 $ $ 7 $ $ 8 $ $ 9 $ 10 11 12
$ a $
0.54 -0.394 -0.216 -0.119 -0.060 -0.022 $ \fbox{0.003}$ 0.021 0.035 0.046
0.56 -0.531 -0.247 -0.093 $\fbox{0.000} $ 0.059 0.100 0.130 0.152 0.168
0.58 -0.906 -0.433 -0.176 -0.021 $\fbox{0.079} $ 0.148 0.197 0.234 0.262
0.60 -1.928 -1.079 -0.618 -0.340 -0.160 -0.036 $ \fbox{0.051}$ 0.117 0.167
[1]

Dominique Zosso, Braxton Osting. A minimal surface criterion for graph partitioning. Inverse Problems and Imaging, 2016, 10 (4) : 1149-1180. doi: 10.3934/ipi.2016036

[2]

Giovanni Bellettini, Matteo Novaga, Giandomenico Orlandi. Eventual regularity for the parabolic minimal surface equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5711-5723. doi: 10.3934/dcds.2015.35.5711

[3]

Ramzi Alsaedi. Perturbation effects for the minimal surface equation with multiple variable exponents. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 139-150. doi: 10.3934/dcdss.2019010

[4]

Jiří Minarčík, Michal Beneš. Minimal surface generating flow for space curves of non-vanishing torsion. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022011

[5]

Joel Andersson, Leo Tzou. Stability for a magnetic Schrödinger operator on a Riemann surface with boundary. Inverse Problems and Imaging, 2018, 12 (1) : 1-28. doi: 10.3934/ipi.2018001

[6]

Lok Ming Lui, Chengfeng Wen, Xianfeng Gu. A conformal approach for surface inpainting. Inverse Problems and Imaging, 2013, 7 (3) : 863-884. doi: 10.3934/ipi.2013.7.863

[7]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615

[8]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1

[9]

Erica Clay, Boris Hasselblatt, Enrique Pujals. Desingularization of surface maps. Electronic Research Announcements, 2017, 24: 1-9. doi: 10.3934/era.2017.24.001

[10]

Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81.

[11]

Olivier Pinaud. Time reversal of surface plasmons. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022106

[12]

Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593

[13]

Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure and Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379

[14]

Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1465-1474. doi: 10.3934/cpaa.2012.11.1465

[15]

Kazuo Aoki, Pierre Charrier, Pierre Degond. A hierarchy of models related to nanoflows and surface diffusion. Kinetic and Related Models, 2011, 4 (1) : 53-85. doi: 10.3934/krm.2011.4.53

[16]

Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112

[17]

Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511

[18]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431

[19]

E.B. Pitman, C.C. Nichita, A.K. Patra, A.C. Bauer, M. Bursik, A. Webb. A model of granular flows over an erodible surface. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 589-599. doi: 10.3934/dcdsb.2003.3.589

[20]

Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1523-1535. doi: 10.3934/cpaa.2012.11.1523

2021 Impact Factor: 1.273

Article outline

Figures and Tables

[Back to Top]