The area formula for $W^{1,n}$-mappings
Malý, Jan
Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994), p. 291-298 / Harvested from Czech Digital Mathematics Library
Access to full text
Full (PDF)

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.

Publié le : 1994-01-01
Classification:  26B15,  26B20,  28A75,  30C65,  46E35 
@article{118668,
     author = {Jan Mal\'y},
     title = {The area formula for $W^{1,n}$-mappings},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {35},
     year = {1994},
     pages = {291-298},
     zbl = {0812.30006},
     mrnumber = {1286576},
     language = {en},
     url = {http://dml.mathdoc.fr/item/118668}
}

Malý, Jan. The area formula for $W^{1,n}$-mappings. Commentationes Mathematicae Universitatis Carolinae, Tome 35 (1994) pp. 291-298. http://gdmltest.u-ga.fr/item/118668/

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. (1983) | MR 0731786

Federer H. Geometric Measure Theory, Springer-Verlag, Grundlehren, 1969. | MR 0257325 | Zbl 0874.49001

Federer H. Surface area II, Trans. Amer. Math. Soc. 55 (1944), 438-456. (1944) | MR 0010611

Feyel D.; De La Pradelle A. Hausdorff measures on the Wiener space, Potential Analysis 1,2 (1992), 177-189. (1992) | MR 1245885 | Zbl 1081.28500

Giaquinta M.; Modica G.; Souček J. Area and the area formula, preprint, 1993. | MR 1293774

Hedberg L.I.; Wolff Th.H. Thin sets in nonlinear potential theory, Ann. Inst. Fourier, Grenoble 33,4 (1983), 161-187. (1983) | MR 0727526 | Zbl 0508.31008

Heinonen J.; Kilpeläinen T.; Martio O. Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Clarendon Press, 1993. | MR 1207810

Malý J. Hölder type quasicontinuity, Potential Analysis 2 (1993), 249-254. (1993) | MR 1245242

Malý J.; Martio O. Lusin's condition (N) and mappings of the class $W^{1,n}$, Preprint 153, University of Jyväskylä, 1992.

Martio O.; Ziemer W.P. Lusin's condition (N) and mappings with non-negative Jacobians, Michigan Math. J., to appear. | MR 1182504

Meyers N.G. Continuity properties of potentials, Duke Math. J. 42 (1975), 157-166. (1975) | MR 0367235 | Zbl 0334.31004

Reshetnyak Yu.G. On the concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Zh. 10 (1969), 1109-1138. (1969) | MR 0276487

Reshetnyak Yu.G. Space Mappings with Bounded Distortion, Transl. Math. Monographs, Amer. Math. Soc., Providence, 1989. | MR 0994644 | Zbl 0667.30018

Ziemer W.P. Weakly Differentiable Functions, Graduate Texts in Mathematics 120, Springer-Verlag, 1989. | MR 1014685 | Zbl 0692.46022