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

TL;DR

This paper demonstrates the existence of bounded solutions to the Allen-Cahn equation in high dimensions whose zero level sets can have any prescribed compact topology, extending the understanding of solution structures.

Contribution

It introduces a method to construct bounded solutions with zero level sets of arbitrary compact topology in R^d for d ≥ 4, using infinite-index solutions.

Findings

Existence of solutions with prescribed compact topology in zero level sets

Construction of solutions with multiple connected components of specified topology

Extension of previous results to higher dimensions and more complex topologies

Abstract

We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface $\Sigma$ of R^d, with $d\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of $\Sigma$. More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.