The co-area formula for Sobolev mappings
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- by Jan Malý, David Swanson and William P. Ziemer PDF
- Trans. Amer. Math. Soc. 355 (2003), 477-492 Request permission
Abstract:
We extend Federer’s co-area formula to mappings $f$ belonging to the Sobolev class $W^{1,p}(\mathbb {R}^n;\mathbb {R}^m)$, $1 \le m < n$, $p>m$, and more generally, to mappings with gradient in the Lorentz space $L^{m,1}(\mathbb {R}^n)$. This is accomplished by showing that the graph of $f$ in $\mathbb {R}^{n+m}$ is a Hausdorff $n$-rectifiable set.References
- David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
- G. V. Badaljan, Generalized quasianalyticity and a uniqueness criterion for a certain class of analytic functions, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 333–373 (Russian). MR 0344480
- Bojarski, B., Hajłasz, P., and Strzelecki, P., Pointwise inequalities for Sobolev functions revisited, Preprint 2000.
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
- Eilenberg, S., On $\varphi$ measures, Ann. Soc. Pol. de Math. 17 (1938), 251–252.
- Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. MR 110078, DOI 10.1090/S0002-9947-1959-0110078-1
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Wendell H. Fleming, Functions whose partial derivatives are measures, Illinois J. Math. 4 (1960), 452–478. MR 130338
- Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are $p$-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139–158. MR 435361, DOI 10.1512/iumj.1972.22.22013
- Irene Fonseca and Jan Malý, Remarks on the determinant in nonlinear elasticity and fracture mechanics, Applied nonlinear analysis, Kluwer/Plenum, New York, 1999, pp. 117–132. MR 1727444
- Fiorenza, A., and Prignet, A., Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data, preprint.
- Hajłasz, P., Sobolev mappings, co-area formula and related topics, Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk, 2000, 227–254.
- Hencl, S. and Malý, J., Mapping of finite distortion: Hausdorff measure of zero sets, To appear in Math. Ann.
- Janne Kauhanen, Pekka Koskela, and Jan Malý, On functions with derivatives in a Lorentz space, Manuscripta Math. 100 (1999), no. 1, 87–101. MR 1714456, DOI 10.1007/s002290050197
- K. A. Hirsch, On skew-groups, Proc. London Math. Soc. 45 (1939), 357–368. MR 0000036, DOI 10.1112/plms/s2-45.1.357
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Jan Malý, Hölder type quasicontinuity, Potential Anal. 2 (1993), no. 3, 249–254. MR 1245242, DOI 10.1007/BF01048508
- Jan Malý, The area formula for $W^{1,n}$-mappings, Comment. Math. Univ. Carolin. 35 (1994), no. 2, 291–298. MR 1286576
- Jan Malý, Absolutely continuous functions of several variables, J. Math. Anal. Appl. 231 (1999), no. 2, 492–508. MR 1669167, DOI 10.1006/jmaa.1998.6246
- Malý, J., Sufficient Conditions for Change of Variables in Integral, Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk (2000) 370–386.
- Malý, J., Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals, Preprint MATH-KMA-2002/74, Charles University, Praha, 2002.
- Jan Malý and Olli Martio, Lusin’s condition (N) and mappings of the class $W^{1,n}$, J. Reine Angew. Math. 458 (1995), 19–36. MR 1310951, DOI 10.1515/crll.1995.458.19
- Malý, J., Swanson, D., and Ziemer, W. P., Fine behavior of functions with gradients in a Lorentz space, In preparation.
- M. Marcus and V. J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. 79 (1973), 790–795. MR 322651, DOI 10.1090/S0002-9904-1973-13319-1
- V. G. Maz′ja and V. P. Havin, A nonlinear potential theory, Uspehi Mat. Nauk 27 (1972), no. 6, 67–138. MR 0409858
- Ju. G. Rešetnjak, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Ž. 10 (1969), 1109–1138 (Russian). MR 0276487
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Swanson, D., Pointwise inequalities and approximation in fractional Sobolev spaces, Studia Math. 149 (2002), 147-174.
- Roberto Van der Putten, On the critical-values lemma and the coarea formula, Boll. Un. Mat. Ital. B (7) 6 (1992), no. 3, 561–578 (Italian, with English summary). MR 1191953
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
Additional Information
- Jan Malý
- Affiliation: Faculty of Mathematics and Physics, Charles University – KMA, Sokolovská 83, 18675 Praha 8, Czech Republic
- Email: maly@karlin.mff.cuni.cz
- David Swanson
- Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
- Email: dswanson@math.tamu.edu
- William P. Ziemer
- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
- Email: ziemer@indiana.edu
- Received by editor(s): December 3, 2001
- Published electronically: August 27, 2002
- Additional Notes: The research of the first author is supported in part by the Research Project MSM 113200007 from the Czech Ministry of Education, Grant No. 201/00/0767 from the Grant Agency of the Czech Republic (GA ČR) and Grant No. 165/99 from the Grant Agency of Charles University (GA UK)
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 355 (2003), 477-492
- MSC (2000): Primary 46E35, 46E30; Secondary 26B10, 26B35, 49Q15
- DOI: https://doi.org/10.1090/S0002-9947-02-03091-X
- MathSciNet review: 1932709