Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Sobolev spaces; change of variables; area formula; Hölder continuity
Sobolev spaces; change of variables; area formula; Hölder continuity
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.
References:
[{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
[{2}] Federer H.: Geometric Measure Theory. Springer-Verlag, Grundlehren, 1969. MR 0257325 | Zbl 0874.49001
[{3}] Federer H.: Surface area II. Trans. Amer. Math. Soc. 55 (1944), 438-456. MR 0010611
[{4}] Feyel D., de la Pradelle A.: Hausdorff measures on the Wiener space. Potential Analysis 1,2 (1992), 177-189. MR 1245885 | Zbl 1081.28500
[{5}] Giaquinta M., Modica G., Souček J.: Area and the area formula. preprint, 1993. MR 1293774
[{6}] Hedberg L.I., Wolff Th.H.: Thin sets in nonlinear potential theory. Ann. Inst. Fourier, Grenoble 33,4 (1983), 161-187. MR 0727526 | Zbl 0508.31008
[{7}] Heinonen J., Kilpeläinen T., Martio O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford Mathematical Monographs, Clarendon Press, 1993. MR 1207810
[{8}] Malý J.: Hölder type quasicontinuity. Potential Analysis 2 (1993), 249-254. MR 1245242
[{9}] Malý J., Martio O.: Lusin's condition (N) and mappings of the class $W^{1,n}$. Preprint 153, University of Jyväskylä, 1992.
[{10}] Martio O., Ziemer W.P.: Lusin's condition (N) and mappings with non-negative Jacobians. Michigan Math. J., to appear. MR 1182504
[{11}] Meyers N.G.: Continuity properties of potentials. Duke Math. J. 42 (1975), 157-166. MR 0367235 | Zbl 0334.31004
[{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
[{13}] Reshetnyak Yu.G.: Space Mappings with Bounded Distortion. Transl. Math. Monographs, Amer. Math. Soc., Providence, 1989. MR 0994644 | Zbl 0667.30018
[{14}] Ziemer W.P.: Weakly Differentiable Functions. Graduate Texts in Mathematics 120, Springer-Verlag, 1989. MR 1014685 | Zbl 0692.46022
Search
Partner of
