Geodesic
Weighted Korn inequalities in paraboloidal domains
Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 751-765.

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A weighted Korn inequality in a domain $\Omega\subset\mathbb R^n$ with paraboloidal exit $\Pi$ to infinity is obtained. Asymptotic sharpness of the inequality is achieved by using different weight factors for the longitudinal (with respect to the axis of $\Pi$) and transversal displacement vector components and by making the weight factors of the derivatives depend on the direction of differentiation. The solvability of the elasticity problem in the energy class (the closure of $C_0^\infty(\overline\Omega)^n$ in the norm generated by the elastic energy functional) is studied; the dimensions of the kernel and the cokerned of the corresponding operator depend on the exponent $s\in(-\infty,1)$ in the “grate of expansion” of the paraboloid $\Pi$. 
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     author = {S. A. Nazarov},
     title = {Weighted {Korn} inequalities in paraboloidal domains},
     journal = {Matemati\v{c}eskie zametki},
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     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a11/}
}
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S. A. Nazarov. Weighted Korn inequalities in paraboloidal domains. Matematičeskie zametki, Tome 62 (1997) no. 5, pp. 751-765. http://geodesic.mathdoc.fr/item/MZM_1997_62_5_a11/

[1] Nazarov S. A., “Neravenstva Korna, asimptoticheski tochnye dlya tonkikh oblastei”, Vestn. SPbGU. Matem., mekh., astron., 1992, no. 8, 19–24 | MR

[2] Kondratev V. A., Oleinik O. A., “O zavisimosti konstant v neravenstve Korna ot parametra, kharakterizuyuschego geometriyu oblasti”, UMN, 44:6 (1989), 157–158 | MR | Zbl

[3] Kondratiev V. A., Oleinik O. A., “Korn's type inequalities for a class of unbounded domains and applications to boundary-value problems in elasticity”, Elasticity. Mathematical methods and application, eds. G. Eason, R. W. Ogden, Horword, Chichester, 1990, 211–233

[4] Kondratiev V. A., Oleinik O. A., “Hardy's and Korn's type inequalities and their applications”, Rend. Mat. Appl. (7), 10 (1990), 641–666 | MR | Zbl

[5] Nazarov S. A., Plamenevsky B. A., Elliptic problems in domains with piecewice smooth boundaries, Walter de Gruyter, Berlin, 1994

[6] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980

[7] Kondratev V. A., Oleinik O. A., “Kraevye zadachi dlya sistemy teorii uprugosti v neogranichennykh oblastyakh. Neravenstvo Korna”, UMN, 43:5 (1988), 55–98 | MR

[8] Oleinik O. A., “Korn's type inequalities and applications to elasticity”, Atti Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur. Mem. (9), Convegno internazionale in memoria di Vito Volterra, Mat. Appl., 92, 1992, 183–209 | MR | Zbl