Some remarks on the classification of global solutions with asymptotically flat level sets

TL;DR

This paper revisits and simplifies the classification of global solutions with flat level sets for semilinear equations, addressing a conjecture related to De Giorgi and providing a self-contained proof.

Contribution

It offers a simplified, self-contained proof of the classification of certain global solutions for semilinear equations, improving upon previous technical approaches.

Findings

Reproved classification of solutions with flat level sets

Simplified technical steps from previous work

Confirmed conjecture of De Giorgi in specific cases

Abstract

We simplify some technical steps from \cite{S1} in which a conjecture of De Giorgi was addressed. For completeness we make the paper self-contained and reprove the classification of certain global bounded solutions for semilinear equations of the type $$\triangle u=W'(u),$$ where $W$ is a double well potential.