| Title:
|
The area formula for $W^{1,n}$-mappings (English) |
| Author:
|
Malý, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
35 |
| Issue:
|
2 |
| Year:
|
1994 |
| Pages:
|
291-298 |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $f$ be a mapping in the Sobolev space $W^{1,n}(\Omega,\bold R^n)$. Then the change of variables, or area formula holds for $f$ provided removing from counting into the multiplicity function the set where $f$ is not approximately Hölder continuous. This exceptional set has Hausdorff dimension zero. (English) |
| Keyword:
|
Sobolev spaces |
| Keyword:
|
change of variables |
| Keyword:
|
area formula |
| Keyword:
|
Hölder continuity |
| MSC:
|
26B15 |
| MSC:
|
26B20 |
| MSC:
|
28A75 |
| MSC:
|
30C65 |
| MSC:
|
46E35 |
| idZBL:
|
Zbl 0812.30006 |
| idMR:
|
MR1286576 |
| . |
| Date available:
|
2009-01-08T18:11:05Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118668 |
| . |
| Reference:
|
[{1}] Bojarski B., Iwaniec T.: Analytical foundations of the theory of quasiconformal mapping in $\bold R^n$.Ann. Acad. Sci. Fenn. Ser. A I. Math. 8 (1983), 257-324. MR 0731786 |
| Reference:
|
[{2}] Federer H.: Geometric Measure Theory.Springer-Verlag, Grundlehren, 1969. Zbl 0874.49001, MR 0257325 |
| Reference:
|
[{3}] Federer H.: Surface area II.Trans. Amer. Math. Soc. 55 (1944), 438-456. MR 0010611 |
| Reference:
|
[{4}] Feyel D., de la Pradelle A.: Hausdorff measures on the Wiener space.Potential Analysis 1,2 (1992), 177-189. Zbl 1081.28500, MR 1245885 |
| Reference:
|
[{5}] Giaquinta M., Modica G., Souček J.: Area and the area formula.preprint, 1993. MR 1293774 |
| Reference:
|
[{6}] Hedberg L.I., Wolff Th.H.: Thin sets in nonlinear potential theory.Ann. Inst. Fourier, Grenoble 33,4 (1983), 161-187. Zbl 0508.31008, MR 0727526 |
| Reference:
|
[{7}] Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations.Oxford Mathematical Monographs, Clarendon Press, 1993. MR 1207810 |
| Reference:
|
[{8}] Malý J.: Hölder type quasicontinuity.Potential Analysis 2 (1993), 249-254. MR 1245242 |
| Reference:
|
[{9}] Malý J., Martio O.: Lusin's condition (N) and mappings of the class $W^{1,n}$.Preprint 153, University of Jyväskylä, 1992. |
| Reference:
|
[{10}] Martio O., Ziemer W.P.: Lusin's condition (N) and mappings with non-negative Jacobians.Michigan Math. J., to appear. MR 1182504 |
| Reference:
|
[{11}] Meyers N.G.: Continuity properties of potentials.Duke Math. J. 42 (1975), 157-166. Zbl 0334.31004, MR 0367235 |
| Reference:
|
[{12}] Reshetnyak Yu.G.: On the concept of capacity in the theory of functions with generalized derivatives.Sibirsk. Mat. Zh. 10 (1969), 1109-1138. MR 0276487 |
| Reference:
|
[{13}] Reshetnyak Yu.G.: Space Mappings with Bounded Distortion.Transl. Math. Monographs, Amer. Math. Soc., Providence, 1989. Zbl 0667.30018, MR 0994644 |
| Reference:
|
[{14}] Ziemer W.P.: Weakly Differentiable Functions.Graduate Texts in Mathematics 120, Springer-Verlag, 1989. Zbl 0692.46022, MR 1014685 |
| . |
