Solutions to the Allen Cahn equation and minimal surfaces

DOI10.1007/s00032-011-0155-6zbMath1272.35104OpenAlexW1984816666MaRDI QIDQ640705

Wei, Juncheng, Manuel A. del Pino

Publication date: 19 October 2011

Published in: Milan Journal of Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00032-011-0155-6

zbMATH Keywords

minimal surfacesDe Giorgi conjectureAllen Cahn equationinfinite dimensional Lyapunov-Schmidt reduction

Mathematics Subject Classification ID

Singular perturbations in context of PDEs (35B25) Minimal surfaces and optimization (49Q05) Minimal surfaces in differential geometry, surfaces with prescribed mean curvature (53A10) Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian (35J91)

Related Items (5)

Gradient estimates for weighted \(p\)-Laplacian equations on Riemannian manifolds with a Sobolev inequality and integral Ricci bounds ⋮ Global minimizers of the Allen-Cahn equation in dimension \(n\geq 8\) ⋮ Bifurcation of solutions to the Allen–Cahn equation ⋮ On a free boundary problem and minimal surfaces ⋮ Clifford Tori and the singularly perturbed Cahn-Hilliard equation

Cites Work

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• Solutions of the Allen-Cahn equation which are invariant under screw-motion
• On the oriented Plateau problem
• Saddle-shaped solutions of bistable diffusion equations in all of \(\mathbb R^{2m}\)
• Stability of translating solutions to mean curvature flow
• Serrin's overdetermined problem and constant mean curvature surfaces
• Flattening results for elliptic PDEs in unbounded domains with applications to overdetermined problems
• A complete embedded minimal surface in \({\mathbb{R}}^ 3\) with genus one and three ends
• Entire solutions of the minimal surface equation
• On a conjecture of De Giorgi and some related problems
• Morse index and Gauss maps of complete minimal surfaces in Euclidean 3- space
• Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle
• A counterexample to a conjecture by De Giorgi in large dimensions
• Regularity of flat level sets in phase transitions
• Towards a counter-example to a conjecture of De Giorgi in high dimensions
• Some interior regularity theorems for minimal surfaces and an extension of Bernstein's theorem
• Minimal varieties in Riemannian manifolds
• Minimal cones and the Bernstein problem
• Index and nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces
• Entire solutions of semilinear elliptic equations in ℝ³ and a conjecture of De Giorgi
• Stable and finite Morse index solutions on 𝐑ⁿ or on bounded domains with small diffusion

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