Finite Morse index implies finite ends

Abstract

We prove that finite Morse index solutions to the Allen-Cahn equation in ${\mathbb{R}}^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularity of clustering interfaces for the singularly perturbed Allen-Cahn equation in ${\mathbb{R}}^n$. Using an indirect blow-up technique, in the spirit of the classical Colding-Minicozzi theory in minimal surfaces, we show that the {\bf obstruction} to the uniform second order regularity of clustering interfaces in ${\mathbb{R}}^n$ is associated to the existence of nontrivial entire solutions to a (finite or infinite) {\bf Toda system} in ${\mathbb{R}}^{n-1}$. For finite Morse index solutions in ${\mathbb{R}}^2$, we show that this obstruction does not exist by using information on stable solutions of the Toda system.

66 pages

References in corpus (3)

• The structure of finite Morse index solutions to two free boundary problems in ${\mathbb{R}}^2$
• Two-end solutions to the Allen-Cahn equation in ${\mathbb{R}}^3$
• Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in ${\mathbb{R}}^2$

Cited by in corpus (3)

• A note on the Morse index of 2k-ended phase transitions in R^2
• Axially symmetric solutions of Allen-Cahn equation with finite Morse index
• Second order estimates on transition layers