Geodesic
On the existence of a contraction mapping preserving boundary values
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 7 (2007) no. 2, pp. 65-87 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {A. I. Parf\"enov},
     title = {On the existence of a contraction mapping preserving boundary values},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {65--87},
     year = {2007},
     volume = {7},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2007_7_2_a4/}
}
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A. I. Parfënov. On the existence of a contraction mapping preserving boundary values. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 7 (2007) no. 2, pp. 65-87. http://geodesic.mathdoc.fr/item/VNGU_2007_7_2_a4/

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