Allen-Cahn min-max on surfaces. (English) Zbl 1459.35179
Author’s abstract: We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, \(2\)-dimensional Riemannian manifolds, as motivated by the work of [M. A. M. Guaraco, J. Differ. Geom. 108, No. 1, 91–133 (2018; Zbl 1387.49060)]. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to [K. Wang and J. Wei, Commun. Pure Appl. Math. 72, No. 5, 1044–1119 (2019; Zbl 1418.35190)], we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding “entire” singularity models constructed by [M. del Pino et al., J. Differ. Geom. 93, No. 1, 67–131 (2013; Zbl 1275.53015)] with Morse index \(1.\) The argument is completed by a Morse index estimate on these singularity models.
Reviewer: Dian K. Palagachev (Bari)
MSC:
| 35J61 | Semilinear elliptic equations |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 35J20 | Variational methods for second-order elliptic equations |
References:
| [1] | W. K. Allard and F. J. Almgren, Jr. The structure of stationary one dimensional varifolds with positive density.Invent. Math.34(1976), no. 2, 83-97, MR0425741, Zbl 0339.49020. · Zbl 0339.49020 |
| [2] | L. Ambrosio and X. Cabr´e. Entire solutions of semilinear elliptic equations inR3and a conjecture of De Giorgi.J. Amer. Math. Soc.13 (2000), no. 4, 725-739, MR1775735, Zbl 0968.35041. · Zbl 0968.35041 |
| [3] | Lipman Bers, Fritz John, and Martin Schechter.Partial differential equations. American Mathematical Society, Providence, R.I., 1979. With supplements by Lars G ˙arding and A. N. Milgram, With a preface by A. S. Householder, Reprint of the 1964 original, Lectures in Applied Mathematics, 3A, MR0598466, Zbl 0514.35001. · Zbl 0514.35001 |
| [4] | O. Chodosh and D. Maximo. On the topology and index of minimal surfaces.J. Differential Geom.104(2016), no. 3, 399-418, MR3568626, Zbl 1357.53016. · Zbl 1357.53016 |
| [5] | E. De Giorgi. Convergence problems for functionals and operators. In Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pages 131-188. Pitagora, Bologna, 1979, MR0533166, Zbl 0405.49001. · Zbl 0405.49001 |
| [6] | M. del Pino, M. Kowalczyk, and F. Pacard. Moduli space theory for the Allen-Cahn equation in the plane.Trans. Amer. Math. Soc.365 (2013), no. 2, 721-766, MR2995371, Zbl 1286.35018. · Zbl 1286.35018 |
| [7] | M. del Pino, M. Kowalczyk, and J. Wei. On De Giorgi’s conjecture in dimensionN≥9.Ann. of Math. (2)174(2011), no. 3, 1485-1569, MR2846486, Zbl 1238.35019. · Zbl 1238.35019 |
| [8] | M. del Pino, M. Kowalczyk, and J. Wei. Entire solutions of the AllenCahn equation and complete embedded minimal surfaces of finite total curvature inR3.J. Differential Geom.93(2013), no. 1, 67-131, MR3019512, Zbl 1275.53015. · Zbl 1275.53015 |
| [9] | M. del Pino, M. Kowalczyk, J. Wei, and J. Yang. Interface foliation near minimal submanifolds in Riemannian manifolds with positive Ricci curvature.Geom. Funct. Anal.20(2010), no. 4, 918-957, MR2729281, Zbl 1213.35219. · Zbl 1213.35219 |
| [10] | H. del Rio Guerra, C. E. Garza-Hume, and P. Padilla. Geodesics, soap bubbles and pattern formation in Riemannian surfaces.J. Geom. Anal. 13(2003), no. 4, 595-604, MR2005155, Zbl 1085.49047. · Zbl 1085.49047 |
| [11] | A. Farina, L. Mari, and E. Valdinoci. Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds. Comm. Partial Differential Equations38(2013), no. 10, 1818-1862, MR3169764, Zbl 1287.58011. · Zbl 1287.58011 |
| [12] | N. Ghoussoub and C. Gui. On a conjecture of De Giorgi and some related problems.Math. Ann.311(1998), no. 3, 481-491, MR1637919, Zbl 0918.35046. · Zbl 0918.35046 |
| [13] | A. Grigor’yan, Y. Netrusov, and S.-T. Yau. Eigenvalues of elliptic operators and geometric applications. InSurveys in Differential Geometry. Vol. IX, volume 9 ofSurv. Differ. Geom., pages 147-217. Int. Press, Somerville, MA, 2004, MR2195408, Zbl 1061.58027. · Zbl 1050.53002 |
| [14] | M. A. M. Guaraco. Min-max for phase transitions and the existence of embedded minimal hypersurfaces.J. Differential Geom.108(2018), no. 1, 91-133, MR3743704, Zbl 1387.49060. · Zbl 1387.49060 |
| [15] | C. Gui. Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions.J. Differential Equations252(2012), no. 11, 5853- 5874, MR2911416, Zbl 1250.35078. · Zbl 1250.35078 |
| [16] | J. E. Hutchinson and Y. Tonegawa. Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory.Calc. Var. Partial Differential Equations10(2000), no. 1, 49-84, MR1803974, Zbl 1070.49026. · Zbl 1070.49026 |
| [17] | M. Kowalczyk, Y. Liu, and F. Pacard. The space of 4-ended solutions to the Allen-Cahn equation in the plane.Ann. Inst. H. Poincar´e Anal. Non Lin´eaire29(2012), no. 5, 761-781, MR2971030, Zbl 1254.35219. · Zbl 1254.35219 |
| [18] | L. Lyusternik and L. Schnirelmman. Topological methods in variational problems and their application to the differential geometry of surfaces.Uspehi Matem. Nauk (N.S.)2(1947), no. 1(17), 166-217, MR0029532. · Zbl 1446.53052 |
| [19] | Y. Liu and J. Wei. A complete classification of finite Morse index solutions to elliptic sine-Gordon equation in the plane, arXiv preprint (2018) arXiv:1806.06921. |
| [20] | Y. Liu, K. Wang, and J. Wei. Global minimizers of the Allen-Cahn equation in dimensionn≥8.J. Math. Pures Appl. (9)108(2017), no. 6, 818-840, MR3723158, Zbl 1380.35071. · Zbl 1380.35071 |
| [21] | C. Mantoulidis.Geometric variational problems in mathematical physics. PhD thesis, Stanford University, June 2017. · Zbl 1375.58017 |
| [22] | F. Pacard and J. Wei. Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones.J. Funct. Anal.264(2013), no. 5, 1131-1167, MR3010017, Zbl 1281.35046. · Zbl 1281.35046 |
| [23] | O. Savin. Regularity of flat level sets in phase transitions.Ann. of Math. (2)169(2009), no. 1, 41-78, MR2480601, Zbl 1180.35499. · Zbl 1180.35499 |
| [24] | L. Simon.Lectures on geometric measure theory, volume 3 ofProceedings of the Centre for Mathematical Analysis, Australian National University. Australian National University, Centre for Mathematical Analysis, Canberra, 1983, MR0756417, Zbl 0546.49019. · Zbl 0546.49019 |
| [25] | R. Schoen and L. Simon. Regularity of stable minimal hypersurfaces. Comm. Pure Appl. Math.34(1981), no. 6, 741-797, MR0634285, Zbl 0497.49034. · Zbl 0497.49034 |
| [26] | Y. Tonegawa. On stable critical points for a singular perturbation problem.Comm. Anal. Geom.13(2005), no. 2, 439-459, MR2154826, Zbl 1105.35008. · Zbl 1105.35008 |
| [27] | Y. Tonegawa and N. Wickramasekera. Stable phase interfaces in the van der Waals-Cahn-Hilliard theory.J. Reine Angew. Math.668 (2012), 191-210, MR2948876, Zbl 1244.49077. · Zbl 1244.49077 |
| [28] | K. Wang. A new proof of Savin’s theorem on Allen-Cahn equations.J. Eur. Math. Soc. (JEMS)19(2017), no. 10, 2997-3051, MR3713000, Zbl 1388.35069. · Zbl 1388.35069 |
| [29] | K. Wang. Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation inR2.NoDEA Nonlinear Differential Equations Appl.24(2017), no. 5, Art. 58, 17 pp., MR3690656, Zbl 1382.35109. · Zbl 1382.35109 |
| [30] | K. Wang and J. Wei. Finite Morse index implies finite ends.Comm. Pure Appl. Math.72(2019), no. 5, 1044-1119, MR3935478, Zbl 1418.35190. · Zbl 1418.35190 |
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