From Constant mean Curvature Hypersurfaces to the Gradient Theory of Phase Transitions

By Frank Pacard, Manuel Ritor'e. 2003

176 citations recorded by the source corpus.

We construct solutions of the Cahn-Hilliard equation whose nodal set converges to a given constant mean curvature hypersurface in a Riemannian manifold.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4dcd52c76a1ba11c0fb07b87e851e49437800e4b

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In Applied Mathematics

1. Jun-cheng Wei
2. Manuel del Pino
3. M. Kowalczyk
4. Frank Pacard
5. W. Yao
6. Yanyan Li
7. Hao Xu
8. P. Gaspar

In Geometry and Topology

1. Jun-cheng Wei
2. Manuel del Pino
3. P. Gaspar
4. M. Kowalczyk
5. Yanyan Li
6. Hao Xu
7. W. Yao
8. Rafe Mazzeo

Citations in the corpus

• The Allen–Cahn equation on closed manifolds (2016, result): “This approach to Almgren–Pitts theorem can be interpreted as a converse of the results of Pacard and Ritoré [48].”
• MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2 (2010, methodology): “The proof of this fact follows the method first introduced in [28].”
• Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature (2010, background): “The nature of these solutions is drastically different from the single-interface solution by Pacard and Ritoré[28].”
• Intermediate reduction method and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials (2015, methodology): “Treating the discrete problem needs a method, technically delicate, which we interpret as an intermediate procedure between the finite and the infinite dimensional one (see for instance [22], [21], [38] and references therein for the latter).”
• Entire solutions of the Allen-Cahn equation and complete embedded minimal surfaces. (2009, methodology): “It is also the basic element, in a compact-manifold version, to build solutions to the small-parameter Allen-Cahn equation in Pacard and Ritor¶e [ 32 ].”
• Serrin’s overdetermined problem and constant mean curvature surfaces (2013, background): “[19] F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom.”
• Multi-bump solutions of -Δn = K(x)u (n+2)/(n-2) on lattices in ℝ n (2018, methodology): “For compact manifold case, we refer to Del Pino-Kowalczyk-Wei [20, 21] and Pacard-Ritore [45], and for noncompact manifolds, we refer to Del Pino-Kowalczyk-Pacard-Wei [22], Del Pino-Kowalczyk-Wei [19] and the references therein.”
• Concentration on curves for nonlinear Schrödinger Equations (2007, background): “In [1, 20, 19, 33] related singular perturbation problems involving the”
• The Toda System and Clustering Interfaces in the Allen–Cahn equation (2008, background): “In [34], Allen–Cahn in dimensions three or higher in a compact manifold has been considered, establishing that associated to a nondegenerate minimal hypersurface a solution with a single interface exists, in analogy with the result in [24].”
• Foliations by constant mean curvature tubes (2003, background): “For example, the second author and Ritoré [10] prove the existence of positive solutions to ε(2) ∆u− u(3) + u = 0 which concentrate along a minimal submanifold Σ as ε ↘ 0.”