On the existence of a contraction mapping preserving boundary values

Let $\mu$ be a finite positive measure defined in the cube $Q=(0,1)^n$ of Euclidean space. Let $S$ be one of the faces of $Q$. For $mp>n$, we consider the subspace $Z$ of the Sobolev space $W_p^m(Q)$ comprising the functions with the zero total trace on $\partial Q\setminus S$. We investigate whether there exists a nonlinear operator $T$ which is bounded in $Z$, preserves the total trace on $S$, and is contracting in the space $L_{2,\mu}(Q)$. Connections of this condition with the interpolation theory of Banach spaces, indefinite spectral problems, and nonlinear differential equations are presented. We prove some sufficient conditions (in terms of $n$, $m$, $p$, and $\mu$) and the one necessary for the existence of $T$. A criterion (in terms of $\mu$) for the existence of $T$ is obtained when $n=1$. The proof of some of the results employs polynomial approximation of functions with the small Sobolev norm.