Abstract
Let \({(\mathcal {M},\tilde{g})}\) be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation
where \({\varepsilon}\) is a small parameter. Let \({{\mathcal {K} \subset \mathcal {M}}}\) be an (N - 1)-dimensional smooth minimal submanifold that separates \({\mathcal{M}}\) into two disjoint components. Assume that \({\mathcal{K}}\) is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that \({{|A_\mathcal {K} |^2 + {\rm {Ric}}_{\tilde g} (v _{\mathcal K}, v_{\mathcal K})}}\) is positive along \({\mathcal{K}}\). Then for each integer m ≥ 2, we establish the existence of a sequence \({\varepsilon = \varepsilon{_j} \rightarrow 0}\), and solutions \({u_\varepsilon}\) with m-transition layers near \({\mathcal{K}}\), with mutual distance \({O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alikakos N.D., Chen X., Fusco G.: Motion of a droplet by surface tension along the boundary. Cal. Var. PDE 11, 233–306 (2000)
Allen S., Cahn J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metall. 27, 1084–1095 (1979)
I. Birindelli, R. Mazzeo, Symmetry of solution of two-phase semilinear elliptic equations on hyperbolic space, http://arxiv.org/abs/0806.2952v1
Bronsard L., Stoth B.: On the existence of high multiplicity interfaces. Math. Res. Lett. 3, 117–131 (1996)
Caffarelli L., Córdoba A.: Uniform convergence of a singular perturbation problem. Comm. Pure Appl. Math. XLVII, 1–12 (1995)
Chavel I.: Riemannian Geometry-A Modern Introduction, Cambridge Tracts in Math. 108. Cambridge Univ. Press, Cambridge (1993)
Dancer E.N., Yan S.: Multi-layer solutions for an elliptic problem. J. Diff. Eqns. 194, 382–405 (2003)
del Pino M., Kowalczyk M., Wei J.: Concentration on curves for nonlinear Schrödinger equations. Comm. Pure Appl. Math. 70, 113–146 (2007)
del Pino M., Kowalczyk M., Wei J.: The Toda system and clustering interface in the Allen–Cahn equation. Archive Rational Mechanical Analysis 190(1), 141–187 (2008)
del Pino M., Kowalczyk M., Pacard F., Wei J.: Multiple-end solutions to the Allen–Cahn equation in ℝ2, J. Funct. Anal. 258(2), 458–503 (2010)
Garza-Hume C.E., Padilla P.: Closed geodesic on oval surfaces and pattern formation, Comm. Anal. Geom. 11(2), 223–233 (2003)
Kato T.: Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin (1995)
Kohn R.V., Sternberg P.: Local minimizers and singular perturbations. Proc. Royal Soc. Edinburgh 11A, 69–84 (1989)
B.M. Levitan, I.S. Sargsjan, Sturm-Liouville and Dirac Operator. Mathematics and its Application (Soviet Series) 59. Kluwer Academic Publishers Group, Dordrecht (1991).
Li P., Yau S.-T.: On the Schrödinger equation and the eigenvalue problem. Commun. Math. Phys. 88, 309–318 (1983)
Mahmoudi F., Malchiodi A.: Concentration on minimal submanifolds for a singularly perturbed Neumann problem. Adv. Math. 209(2), 460–525 (2007)
Mahmoudi F., Mazzeo R., Pacard F.: Constant mean curvature hypersurfaces condensing on a submanifold. Geom. Funct. Anal. 16(4), 924–958 (2006)
Malchiodi A.: Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains. Geom. Funct. Anal. 15(6), 1162–1222 (2005)
Malchiodi A., Montenegro M.: Boundary concentration phenomena for a singularly perturbed elliptic problem. Commun. Pure Appl. Math. 55, 1507–1568 (2002)
Malchiodi A., Montenegro M.: Multidimensional boundary layers for a singularly perturbed Neumann problem. Duke Math. J. 124(1), 105–143 (2004)
Malchiodi A., Wei J.: Boundary interface for the Allen–Cahn equation. J. Fixed Point Theory Appl. 1(2), 305–336 (2007)
Mazzeo R., Pacard F.: Foliations by constant mean curvature tubes. Comm. Anal. Geom. 13(4), 633–670 (2005)
Minakshisundaram S., Pleijel A.: Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Canad. J. Math. 1, 242–256 (1949)
Modica L.: The gradient theory of phase transitions and the minimal interface criterion. Arch. Rat. Mech. Anal. 98, 357–383 (1987)
L. Modica, Convergence to minimal surfaces problem and global solutions of u = 2(u 3 − u), Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), Pitagora, Bologna, (1979), 223–244.
Nakashima K.: Multi-layered stationary solutions for a spatially inhomogeneous Allen–Cahn equation. J. Diff. Eqns. 191, 234–276 (2003)
Nakashima K., Tanaka K.: Clustering layers and boundary layers in spatially inhomogeneous phase transition problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 20(1), 107–143 (2003)
Pacard F., Ritoré M.: From constant mean curvature hypersurfaces to the gradient theory of phase transitions. J. Diff. Geom. 64, 359–423 (2003)
Padilla P., Tonegawa Y.: On the convergence of stable phase transitions. Comm. Pure Appl. Math. 51, 551–579 (1998)
Röger M., Tonegawa Y.: Convergence of phase-field approximations to the Gibbs-Thomson law. Cal. Var. PDE 32, 111–136 (2008)
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer Verlag, Berlin, 1992.
M. Spivak, A Comprehensive Introduction to Differential Geometry, Second edition, Publish or Perish Inc. Wilmington, Del., (1979).
Tonegawa Y.: Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh Sect. A 132(4), 993–1019 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
del Pino, M., Kowalczyk, M., Wei, J. et al. Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature. Geom. Funct. Anal. 20, 918–957 (2010). https://doi.org/10.1007/s00039-010-0083-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-010-0083-6