Perturbation of zero surfaces

Abstract

It is proved that if a smooth function \(u(x), x\in R^3\), 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.