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chart; coordinate transformation; normal vector; normal derivative; extension theorem; Muckenhoupt weight
Given a domain $\Omega $ of class $C^{k,1}$, $k\in \Bbb N $, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of $\Omega $ in the sense that $(\partial- {\partial x_n})\alpha (x',0)= - N(x')$ and that still is of class $C^{k,1}$. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to $k$ on domains of class $C^{k,1}$. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.
[1] Abels, H.: Reduced and generalized Stokes resolvent equations in asymptotically flat layers. {II}: $H_\infty$-calculus. J. Math. Fluid Mech. 7 223-260 (2005). DOI 10.1007/s00021-004-0117-7 | MR 2177128 | Zbl 1083.35085
[2] Chua, S.-K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41 1027-1076 (1992). DOI 10.1512/iumj.1992.41.41053 | MR 1206339 | Zbl 0767.46025
[3] Curbera, G. P., García-Cuerva, J., Martell, J. M., Pérez, C.: Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv. Math. 203 256-318 (2006). DOI 10.1016/j.aim.2005.04.009 | MR 2231047
[4] Evans, L. C.: Partial Differential Equations. American Mathematical Society, Providence (1998). MR 1625845 | Zbl 0902.35002
[5] Farwig, R., Galdi, G. P., Sohr, H.: Very weak solutions of stationary and instationary Navier-Stokes equations with nonhomogeneous data. Nonlinear Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser 64 113-136 (2005). DOI 10.1007/3-7643-7385-7_7 | MR 2185213
[6] Farwig, R., Sohr, H.: Weighted {$L^q$}-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan 49 251-288 (1997). DOI 10.2969/jmsj/04920251 | MR 1601373
[7] Fröhlich, A.: Stokes- und Navier-Stokes-Gleichungen in gewichteten Funktionenr�umen. Shaker Verlag, Aachen (2001).
[8] Fröhlich, A.: The Stokes operator in weighted {$L^q$}-spaces I: Weighted estimates for the Stokes resolvent problem in a half space. J. Math. Fluid Mech. 5 166-199 (2003). MR 1982327
[9] Fröhlich, A.: The Stokes operator in weighted {$L^q$}-spaces {II}: Weighted resolvent estimates and maximal {$L^p$}-regularity. Math. Ann. 339 287-316 (2007). DOI 10.1007/s00208-007-0114-2 | MR 2324721
[10] Galdi, G. P., Simader, C. G., Sohr, H.: A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-\frac1q,q}$. Math. Ann. 331 41-74 (2005). DOI 10.1007/s00208-004-0573-7 | MR 2107439 | Zbl 1064.35133
[11] García-Cuerva, J., Francia, J. L. Rubio de: Weighted norm inequalities and related topics. North Holland, Amsterdam (1985). MR 0807149
[12] Giga, Y.: Analyticity of the semigroup generated by the Stokes operator in $L_r$-spaces. Math. Z. 178 297-329 (1981). DOI 10.1007/BF01214869 | MR 0635201 | Zbl 0473.35064
[13] Nečas, J.: Les Méthodes Directes en Théorie des Équations Elliptiques. Academia, Prague (1967). MR 0227584
[14] Schumacher, K.: Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces. Ann. dell'Univ. di Ferrara 54 123-144 (2008). DOI 10.1007/s11565-008-0038-0 | MR 2403378 | Zbl 1179.35225
[15] Slobodeckiǐ, L. N.: Generalized Sobolev spaces and their application to boundary problems for partial differential equations. Leningrad. Gos. Ped. Inst. Učen. Zap. 197 54-112 (1958). MR 0203222
[16] Stein, E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series. 43, Princeton University Press, Princeton, N.J. (1993). MR 1232192 | Zbl 0821.42001
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