Perturbation of zero surfaces

TL;DR

This paper proves that under certain smoothness and normal derivative conditions, a small perturbation of a function preserves the existence of a closed smooth zero surface.

Contribution

It establishes a rigorous result showing the stability of zero surfaces under small smooth perturbations of the function.

Findings

Small perturbations preserve the existence of closed smooth zero surfaces.

The zero surface's smoothness is maintained under perturbation.

The result applies to functions with positive normal derivative bounds.

Abstract

It is proved that if a smooth function $u(x)$, $x\in \mathbb{R}^3$, such that $\inf_{s\in S}|u_N(s)|>0$, where $u_N$ is the normal derivative of $u$ on $S$, has a closed smooth surface $S$ of zeros, then the function $u(x)+\epsilon v(x)$ has also a closed smooth surface $S_\epsilon$ of zeros. Here $v$ is a smooth function and $\epsilon>0$ is a sufficiently small number.