A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Katrin Schumacher

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 637-648
  • ISSN: 0011-4642

Abstract

top
Given a domain Ω of class C k , 1 , k ∈ ℕ , we construct a chart that maps normals to the boundary of the half space to normals to the boundary of Ω in the sense that ( ∂ - ∂ x n ) α ( 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.

How to cite

top

Schumacher, Katrin. "A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces." Czechoslovak Mathematical Journal 59.3 (2009): 637-648. <http://eudml.org/doc/37948>.

@article{Schumacher2009,
abstract = {Given a domain $\Omega $ of class $C^\{k,1\}$, $k\in \mathbb \{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 $(\partial - \{\partial x_n\})\alpha (x^\{\prime \},0)= - N(x^\{\prime \})$ 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.},
author = {Schumacher, Katrin},
journal = {Czechoslovak Mathematical Journal},
keywords = {chart; coordinate transformation; normal vector; normal derivative; extension theorem; Muckenhoupt weight; chart; coordinate transformation; normal vector; normal derivative; extension theorem; Muckenhoupt weight},
language = {eng},
number = {3},
pages = {637-648},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces},
url = {http://eudml.org/doc/37948},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Schumacher, Katrin
TI - A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 637
EP - 648
AB - Given a domain $\Omega $ of class $C^{k,1}$, $k\in \mathbb {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 $(\partial - {\partial x_n})\alpha (x^{\prime },0)= - N(x^{\prime })$ 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.
LA - eng
KW - chart; coordinate transformation; normal vector; normal derivative; extension theorem; Muckenhoupt weight; chart; coordinate transformation; normal vector; normal derivative; extension theorem; Muckenhoupt weight
UR - http://eudml.org/doc/37948
ER -

References

top
  1. Abels, H., 10.1007/s00021-004-0117-7, J. Math. Fluid Mech. 7 223-260 (2005). (2005) Zbl1083.35085MR2177128DOI10.1007/s00021-004-0117-7
  2. Chua, S.-K., 10.1512/iumj.1992.41.41053, Indiana Univ. Math. J. 41 1027-1076 (1992). (1992) Zbl0767.46025MR1206339DOI10.1512/iumj.1992.41.41053
  3. Curbera, G. P., García-Cuerva, J., Martell, J. M., Pérez, C., 10.1016/j.aim.2005.04.009, Adv. Math. 203 256-318 (2006). (2006) MR2231047DOI10.1016/j.aim.2005.04.009
  4. Evans, L. C., Partial Differential Equations, American Mathematical Society, Providence (1998). (1998) Zbl0902.35002MR1625845
  5. Farwig, R., Galdi, G. P., Sohr, H., 10.1007/3-7643-7385-7_7, Nonlinear Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser 64 113-136 (2005). (2005) MR2185213DOI10.1007/3-7643-7385-7_7
  6. Farwig, R., Sohr, H., 10.2969/jmsj/04920251, J. Math. Soc. Japan 49 251-288 (1997). (1997) MR1601373DOI10.2969/jmsj/04920251
  7. Fröhlich, A., Stokes- und Navier-Stokes-Gleichungen in gewichteten Funktionenr�umen, Shaker Verlag, Aachen (2001). (2001) 
  8. Fröhlich, A., 10.1007/s00021-003-0080-8, J. Math. Fluid Mech. 5 166-199 (2003). (2003) MR1982327DOI10.1007/s00021-003-0080-8
  9. Fröhlich, A., 10.1007/s00208-007-0114-2, Math. Ann. 339 287-316 (2007). (2007) MR2324721DOI10.1007/s00208-007-0114-2
  10. Galdi, G. P., Simader, C. G., Sohr, H., 10.1007/s00208-004-0573-7, Math. Ann. 331 41-74 (2005). (2005) Zbl1064.35133MR2107439DOI10.1007/s00208-004-0573-7
  11. García-Cuerva, J., Francia, J. L. Rubio de, Weighted norm inequalities and related topics, North Holland, Amsterdam (1985). (1985) MR0807149
  12. Giga, Y., 10.1007/BF01214869, Math. Z. 178 297-329 (1981). (1981) Zbl0473.35064MR0635201DOI10.1007/BF01214869
  13. Nečas, J., Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague (1967). (1967) MR0227584
  14. Schumacher, K., 10.1007/s11565-008-0038-0, Ann. dell'Univ. di Ferrara 54 123-144 (2008). (2008) Zbl1179.35225MR2403378DOI10.1007/s11565-008-0038-0
  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). (1958) MR0203222
  16. Stein, E., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. 43, Princeton University Press, Princeton, N.J. (1993). (1993) Zbl0821.42001MR1232192

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.