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 
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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     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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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Azizov T. Ya., Iokhvidov I. S., Osnovy teorii lineinykh operatorov v prostranstvakh s indefinitnoi metrikoi, Nauka, M., 1986 | MR

[2] Berg I., Lefstrem I., Interpolyatsionnye prostranstva. Vvedenie, Mir, M., 1980 | MR

[3] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Editorial URSS, M., 2002

[4] Mazya V. G., Prostranstva S. L. Soboleva, Leningr. un-t, L., 1985 | MR | Zbl

[5] Parfenov A. I., “Ob odnom kriterii vlozheniya interpolyatsionnykh prostranstv i ego prilozhenii k indefinitnym spektralnym zadacham”, Sib. mat. zhurn., 44:4 (2003), 810–819 | MR | Zbl

[6] Parfenov A. I., “Ob uslovii Churgusa v indefinitnykh zadachakh Shturma–Liuvillya”, Mat. trudy, 7:1 (2004), 153–188 | MR | Zbl

[7] Parfenov A. I., Szhimayuschii operator i granichnye znacheniya, Preprint No 155, Institut matematiki SO RAN, Novosibirsk, 2005

[8] Gajda Z., “Local stability of the functional equation characterizing polynomial functions”, Ann. Pol. Math., 52:2 (1990), 119–137 | MR | Zbl

[9] Ivanov S., Kalton N., “Interpolation of subspaces and applications to exponential bases”, Algebra i analiz, 13:2 (2001), 93–115 | MR

[10] Yomdin Y., “The set of zeroes of an “almost polynomial” function”, Proc. Amer. Math. Soc., 90:4 (1984), 538–542 | MR | Zbl