The co-area formula for Sobolev mappings

Transactions of the American Mathematical Society

Volumn 355, Issue 2, 2003, Pages 477-492

  Malý, Jan ^a   Swanson, David ^b   Ziemer, William P ^c  

^a CHARLES UNIVERSITY   (Czech Republic)

^b TEXAS A AND M UNIVERSITY   (United States)

^c INDIANA UNIVERSITY   (United States)

Author keywords

Area formula; Co area formula; Orlicz space; Rectifiability; Sobolev mapping

Indexed keywords

EID: 0037319535     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-02-03091-x     Document Type: Article

References (33)

1. [AH] Adams, D. R. and Hedberg, L. I., Function Spaces and Potential Theory, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1996. MR 97j:46024
2. [BZ] Bagby, T. and Ziemer, W. P., Pointwise differentiability and absolute continuity, Trans. Amer. Math. Soc. 191 (1974), 129-148. MR 49:9219
3. [BHS] Bojarski, B., Hajłasz, P., and Strzelecki, P., Pointwise inequalities for Sobolev functions revisited, Preprint 2000.
4. [C] Cesari, L., Sulle trasformazioni continue, Ann. Mat. Pura Appl. 21 (1942), 157-188. MR 6:43c
5. [E] Eilenberg, S., On φ measures, Ann. Soc. Pol. de Math. 17 (1938), 251-252.
6. [F1] Federer, H., Surface area (II), Trans. Amer. Math. Soc. 55 (1944), 438-456. MR 6:45a
7. [F2] Federer, H., The (φ,k) rectifiable subsets of n space, Trans. Amer. Math. Soc. 62 (1947), 114-192. MR 9:231c
8. [F3] Federer, H., Some integralgeometric theorems, Trans. Amer. Math. Soc. 77 (1954), 238-261. MR 16:163b
9. [F4] Federer, H., Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418-491. MR 22:961
10. [F5] Federer, H., Geometric Measure Theory, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, New York, Heidelberg, 1969. MR 41:1976
11. [Fl] Fleming, W. H., Functions whose partial derivatives are measures, Illinois J. Math. 4 (1960). 452-478. MR 24:A202
12. [FZ] Federer, H. and Ziemer, W. P., The Lebesgue set of a function whose distribution derivatives are p-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139-158. MR 55:8321
13. [FM] Fonseca, I. and Malý, J., Remarks on the Determinant in Nonlinear Elasticity and Fracture Mechanics, Applied Nonlinear Analysis, Eds. A, Sequiera, H. B. da Veiga and J.H. Videman, Kluwer Academic / Plenum Publishers, New York, 1999, 117-132. MR 2000k:74010
14. [FP] Fiorenza, A., and Prignet, A., Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data, preprint.
15. [H] Hajłasz, P., Sobolev mappings, co-area formula and related topics, Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk, 2000, 227-254.
16. [HM] Hencl, S. and Malý, J., Mapping of finite distortion: Hausdorff measure of zero sets, To appear in Math. Ann.
17. [KKM] Kauhanen, J., Koskela, P. and Maly, J., On functions with derivatives in a Lorentz space, Manuscripta Math. 100 (1999), 87-101. MR 2000j:46064
18. [K] Kuratowski, K., Topology, Vols. I and II, Academic Press, New York, 1966. MR 36:840; MR 41:4467
19. [M] Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics 44, Cambridge University Press, 1995. MR 96h:28006
20. [M1] Malý, J., Holder type quasicontinuity, Potential Analysis 2 (1993), 249-254. MR 941:31007
21. 1,n -mappings, Comment. Math. Univ. Carolinae 85 (1994) 291-298. MR 95h:28007
22. [M3] Malý, J., Absolutely continuous functions of several variables, J. Math. Anal. Appl. 231 (1999) 492-508. MR 2000a:49078
23. [M4] Malý, J., Sufficient Conditions for Change of Variables in Integral, Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk (2000) 370-386.
24. [M5] Malý, J., Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals, Preprint MATH-KMA-2002/74, Charles University, Praha, 2002.
25. 1,n, J. reine angew. Math. 458 (1995) 19-36. MR 95m:26024
26. [MSZ] Malý, J., Swanson, D., and Ziemer, W. P., Fine behavior of functions with gradients in a Lorentz space, In preparation.
27. [MMi] Marcus, M. and Mizel, V. J., Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79 no. 4 (1973) 790-795. MR 48:1013
28. [MH] Maz'ya, V. G. and Havin, V. P., Nonlinear potential theory, Uspekhi Mat. Nauk 27 (1972) 67-138. English translation: Russian Math. Surveys 27 (1972), 71-148. MR 53:13610
29. [R] Reshetnyak, Yu. G., On the concept of capacity in the theory of functions with generalized derivatives (Russian), Sibirsk. Mat. Zh. 10 (1969) 1109-1138. English translation: Siberian Math. J. 10 (1969), 818-842. MR 43:2234
30. [RR] Rado, T. and Reichelderfer, P. V., Continuous Transformations in Analysis, Springer-Verlag, Berlin (1955). MR 18:115c
31. [S] Swanson, D., Pointwise inequalities and approximation in fractional Sobolev spaces, Studia Math. 149 (2002), 147-174.
32. [VP] Van der Putten, R., On the critical-values lemma and the coarea formula (Italian) Boll. Unione Mat. Ital. Sez. B. (7) 6-B (1992) 561-578. MR 93j:49043
33. [Z] Ziemer, W. P., Weakly differentiate functions, Graduate Texts in Mathematics 120, Springer-Verlag, New York, 1989. MR 91e:46046