[Résolubilité normale des problèmes elliptiques linéaires]
L'article est consacré aux problèmes elliptiques générals. Nous considérons des domaines non bornés et les espaces de Hölder. On définit les problèmes limites à l'infinie ce qui permet de formuler la condition nécessaire et suffisante de la résolubilité normale. Nous étudions la structure de l'espace dual et décrivons le sous-espace de fonctionnelles qui déterminent les conditions de resolubilité. Cela nous permet de démontrer que pour les opérateurs de Fredholm, tous les problèmes limites sont inversibles.
The paper is devoted to general linear elliptic problems in Hölder spaces. We consider unbounded domains and define limiting problems at infinity. We give a necessary and sufficient condition of normal solvability through uniqueness of solutions of limiting problems. We study a structure of spaces dual to Hölder spaces and specify the subspace of functionals, which provide the condition of normal solvability. This allows us to prove that for Fredholm operators all limiting operators are invertible.
Publié le :
@article{CRMATH_2002__334_6_457_0,
author = {Volpert, Vitaly and Volpert, Aizik},
title = {Normal solvability of linear elliptic problems},
journal = {Comptes Rendus. Math\'ematique},
pages = {457--462},
publisher = {Elsevier},
volume = {334},
number = {6},
year = {2002},
doi = {10.1016/S1631-073X(02)02286-0},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(02)02286-0/}
}
TY - JOUR AU - Volpert, Vitaly AU - Volpert, Aizik TI - Normal solvability of linear elliptic problems JO - Comptes Rendus. Mathématique PY - 2002 SP - 457 EP - 462 VL - 334 IS - 6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(02)02286-0/ DO - 10.1016/S1631-073X(02)02286-0 LA - en ID - CRMATH_2002__334_6_457_0 ER -
%0 Journal Article %A Volpert, Vitaly %A Volpert, Aizik %T Normal solvability of linear elliptic problems %J Comptes Rendus. Mathématique %D 2002 %P 457-462 %V 334 %N 6 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(02)02286-0/ %R 10.1016/S1631-073X(02)02286-0 %G en %F CRMATH_2002__334_6_457_0
Volpert, Vitaly; Volpert, Aizik. Normal solvability of linear elliptic problems. Comptes Rendus. Mathématique, Tome 334 (2002) no. 6, pp. 457-462. doi : 10.1016/S1631-073X(02)02286-0. http://www.numdam.org/articles/10.1016/S1631-073X(02)02286-0/
[1] Estimates near the boundary for solutions of elliptic partial diffrential equations satisfying general boundary conditions, Comm. Pure Appl. Math., Volume 12 (1959), pp. 623-727 17 (1964) 35–92
[2] Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., Volume 8 (1955), pp. 503-538
[3] Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obshch., Volume 16 (1967), pp. 209-292 English transl.: Trans. Math. Soc. 16 (1967)
[4] Fredholm property of a class of elliptic operators on non-compact manifolds, Duke Math. J., Volume 48 (1981), pp. 289-312
[5] Normal solvability and the noethericity of elliptic operators in spaces of functions on , Part I, Zap. Nauch Sem. LOMI, Volume 110 (1981), pp. 120-140 English transl.: J. Soviet Math. 25 (1) (1984) 884–901
[6] On the Fredholm property of the Stokes operator in a layer-like domain, J. Anal. Appl., Volume 20 (2001) no. 1, pp. 155-182
[7] The Fredholm property of general boundary value problems on noncompact manifolds, and limit operators, Dokl. Akad. Nauk, Volume 325 (1992) no. 2, pp. 237-241 (in Russian). English transl.: Russian Acad. Sci. Dokl. Math. 46 (1) (1993) 53–58
[8] Topological degree for elliptic operators in unbounded cylinders, Adv. Differential Equations, Volume 4 (1999) no. 6, pp. 777-812
[9] A Fredholm theory for a class of first-order elliptic partial differential operators in , Trans. Amer. Math. Soc., Volume 165 (1972), pp. 75-86
Cité par Sources :
