doi: 10.3934/dcds.2021077

A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing

Department of Mathematics, UC Irvine, 340 Rowland Hall, Irvine, CA 92697-3875, USA

* Corresponding author: Connor Mooney

Received  February 2021 Revised  March 2021 Published  April 2021

Fund Project: The first author is supported by NSF grant DMS-1854788

We give a proof by foliation that the cones over $ \mathbb{S}^k \times \mathbb{S}^l $ minimize parametric elliptic functionals for each $ k, \, l \geq 1 $. We also analyze the behavior at infinity of the leaves in the foliations. This analysis motivates conjectures related to the existence and growth rates of nonlinear entire solutions to equations of minimal surface type that arise in the study of such functionals.

Citation: Connor Mooney, Yang Yang. A proof by foliation that lawson's cones are $ A_{\Phi} $-minimizing. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021077
References:
[1]

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

[2]

F. J. Almgren Jr.R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., 139 (1977), 217-265.  doi: 10.1007/BF02392238.  Google Scholar

[3]

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

[4]

A. Davini, On calibrations for Lawson's cones, Rend. Sem. Mat. Univ. Padova, 11 (2004), 55-70.   Google Scholar

[5]

E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85.   Google Scholar

[6]

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

[7]

G. De Philippis and E. Paolini, A short proof of the minimality of the Simons cone, Rend. Sem. Mat. Univ. Padova, 121 (2009), 233-241.  doi: 10.4171/RSMUP/121-14.  Google Scholar

[8]

M. DelgadinoF. MaggiC. 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

[9]

K. Ecker and G. Huisken, A Bernstein result for minimal graphs of controlled growth, J. Differential Geom., 31 (1990), 397-400.   Google Scholar

[10]

W. Fleming, On the oriented Plateau problem, Rend. Circolo Mat. Palermo, 9 (1962), 69-89.  doi: 10.1007/BF02849427.  Google Scholar

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

H. Jenkins, On $2$-dimensional variational problems in parametric form, Arch. Ration. Mech. Anal., 8 (1961), 181-206.  doi: 10.1007/BF00277437.  Google Scholar

[13]

H. B. Lawson Jr., The equivariant Plateau problem and interior regularity, Trans. Amer. Math. Soc., 173 (1972), 231-249.  doi: 10.1090/S0002-9947-1972-0308905-4.  Google Scholar

[14]

C. Mooney, Entire solutions to equations of minimal surface type in six dimensions, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.2307/1969841.  Google Scholar

[15]

F. Morgan, The cone over the Clifford torus in $\mathbb{R}^4$ is $\Phi$-minimizing, Math. Ann., 289 (1991), 341-354.  doi: 10.1007/BF01446576.  Google Scholar

[16]

P. Simoes, A Class of Minimal Cones in $\mathbb{R}^n, n \geq 8$ that Minimize Area, Ph.D thesis, UC Berkeley, 1973.  Google Scholar

[17]

L. Simon, Entire solutions of the minimal surface equation, J. Differential Geom., 30 (1989), 643-688.   Google Scholar

[18]

L. Simon, On some extensions of Bernstein's theorem, Math. Z., 154 (1977), 265-273.  doi: 10.1007/BF01214329.  Google Scholar

[19]

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

[20]

B. White, The mathematics of F. J. Almgren, Jr., J. Geom. Anal., 8 (1998), 681-702.  doi: 10.1007/BF02922665.  Google Scholar

show all references

References:
[1]

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

[2]

F. J. Almgren Jr.R. Schoen and L. Simon, Regularity and singularity estimates on hypersurfaces minimizing elliptic variational integrals, Acta Math., 139 (1977), 217-265.  doi: 10.1007/BF02392238.  Google Scholar

[3]

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

[4]

A. Davini, On calibrations for Lawson's cones, Rend. Sem. Mat. Univ. Padova, 11 (2004), 55-70.   Google Scholar

[5]

E. De Giorgi, Una estensione del teorema di Bernstein, Ann. Scuola Norm. Sup. Pisa, 19 (1965), 79-85.   Google Scholar

[6]

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

[7]

G. De Philippis and E. Paolini, A short proof of the minimality of the Simons cone, Rend. Sem. Mat. Univ. Padova, 121 (2009), 233-241.  doi: 10.4171/RSMUP/121-14.  Google Scholar

[8]

M. DelgadinoF. MaggiC. 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

[9]

K. Ecker and G. Huisken, A Bernstein result for minimal graphs of controlled growth, J. Differential Geom., 31 (1990), 397-400.   Google Scholar

[10]

W. Fleming, On the oriented Plateau problem, Rend. Circolo Mat. Palermo, 9 (1962), 69-89.  doi: 10.1007/BF02849427.  Google Scholar

[11]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2$^{nd}$ edition, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[12]

H. Jenkins, On $2$-dimensional variational problems in parametric form, Arch. Ration. Mech. Anal., 8 (1961), 181-206.  doi: 10.1007/BF00277437.  Google Scholar

[13]

H. B. Lawson Jr., The equivariant Plateau problem and interior regularity, Trans. Amer. Math. Soc., 173 (1972), 231-249.  doi: 10.1090/S0002-9947-1972-0308905-4.  Google Scholar

[14]

C. Mooney, Entire solutions to equations of minimal surface type in six dimensions, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.2307/1969841.  Google Scholar

[15]

F. Morgan, The cone over the Clifford torus in $\mathbb{R}^4$ is $\Phi$-minimizing, Math. Ann., 289 (1991), 341-354.  doi: 10.1007/BF01446576.  Google Scholar

[16]

P. Simoes, A Class of Minimal Cones in $\mathbb{R}^n, n \geq 8$ that Minimize Area, Ph.D thesis, UC Berkeley, 1973.  Google Scholar

[17]

L. Simon, Entire solutions of the minimal surface equation, J. Differential Geom., 30 (1989), 643-688.   Google Scholar

[18]

L. Simon, On some extensions of Bernstein's theorem, Math. Z., 154 (1977), 265-273.  doi: 10.1007/BF01214329.  Google Scholar

[19]

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

[20]

B. White, The mathematics of F. J. Almgren, Jr., J. Geom. Anal., 8 (1998), 681-702.  doi: 10.1007/BF02922665.  Google Scholar

Figure 1.  The dilations of $ \Sigma_{kl} $ foliate one side of $ C_{kl} $
Figure 2.  The solution curve is contained in the region $ R $ bounded by $ \Gamma_1, \, \Gamma_2 $ and $ \Gamma_3 $
[1]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363

[2]

Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142

[3]

Zongming Guo, Xuefei Bai. On the global branch of positive radial solutions of an elliptic problem with singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1091-1107. doi: 10.3934/cpaa.2008.7.1091

[4]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053

[5]

Tokushi Sato, Tatsuya Watanabe. Singular positive solutions for a fourth order elliptic problem in $R$. Communications on Pure & Applied Analysis, 2011, 10 (1) : 245-268. doi: 10.3934/cpaa.2011.10.245

[6]

Baishun Lai, Qing Luo. Regularity of the extremal solution for a fourth-order elliptic problem with singular nonlinearity. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 227-241. doi: 10.3934/dcds.2011.30.227

[7]

Gisella Croce. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 507-530. doi: 10.3934/dcdss.2012.5.507

[8]

Anna Maria Candela, Addolorata Salvatore. Existence of minimizers for some quasilinear elliptic problems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3335-3345. doi: 10.3934/dcdss.2020241

[9]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102

[10]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. Uniqueness and sign properties of minimizers in a quasilinear indefinite problem. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/10.3934/cpaa.2021078

[11]

Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. Uniqueness and sign properties of minimizers in a quasilinear indefinite problem. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2313-2322. doi: 10.3934/cpaa.2021078

[12]

Zongming Guo, Juncheng Wei. Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Communications on Pure & Applied Analysis, 2008, 7 (4) : 765-786. doi: 10.3934/cpaa.2008.7.765

[13]

Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308

[14]

Florian Krügel. Some properties of minimizers of a variational problem involving the total variation functional. Communications on Pure & Applied Analysis, 2015, 14 (1) : 341-360. doi: 10.3934/cpaa.2015.14.341

[15]

Salvador Villegas. Nonexistence of nonconstant global minimizers with limit at $\infty$ of semilinear elliptic equations in all of $R^N$. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1817-1821. doi: 10.3934/cpaa.2011.10.1817

[16]

Annamaria Canino, Luigi Montoro, Berardino Sciunzi. The jumping problem for nonlocal singular problems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6747-6760. doi: 10.3934/dcds.2019293

[17]

Haiyun Deng, Hairong Liu, Long Tian. Classification of singular sets of solutions to elliptic equations. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2949-2964. doi: 10.3934/cpaa.2020129

[18]

Prashanta Garain, Tuhina Mukherjee. Quasilinear nonlocal elliptic problems with variable singular exponent. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5059-5075. doi: 10.3934/cpaa.2020226

[19]

Michel Chipot, Senoussi Guesmia. On the asymptotic behavior of elliptic, anisotropic singular perturbations problems. Communications on Pure & Applied Analysis, 2009, 8 (1) : 179-193. doi: 10.3934/cpaa.2009.8.179

[20]

Andrzej Szulkin, Shoyeb Waliullah. Infinitely many solutions for some singular elliptic problems. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 321-333. doi: 10.3934/dcds.2013.33.321

2019 Impact Factor: 1.338

Article outline

Figures and Tables

[Back to Top]