Perturbation of zero surfaces

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.