Bounded solutions to the Allen-Cahn equation with level sets of any\n compact topology

2015/06/02 by Alberto Enciso, Daniel Peralta‐Salas, Enciso, Alberto +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1506.00940

openalex publication_date 2015/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We make use of the flexibility of infinite-index solutions to the Allen-Cahn\nequation to show that, given any compact hypersurface \Σ of R^d, with\nd\≥ 4, there is a bounded entire solution of the Allen-Cahn equation on R^d\nwhose zero level set has a connected component diffeomorphic (and arbitrarily\nclose) to a rescaling of \Σ. More generally, we prove the existence of\nsolutions with a finite number of compact connected components of prescribed\ntopology in their zero level sets.\n

Cited by

• Capacity estimates and improved lower bounds for the inner radius of nodal domains

Related

• Catenoidal layers for the Allen-Cahn equation in bounded domains
• Flat level sets of Allen-Cahn equation in half-space
• Allen-Cahn equation and degenerate minimal hypersurface
• Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones
• Lawson cones and the Allen-Cahn equation