## The co-area formula for Sobolev mappings.(English)Zbl 1034.46032

The co-area formula $\int\limits_\Omega g(x)| J_mf(x)| \, dx = \int_{\mathbb{R}^m}\int_{f^{-1}(y)}g(x) \mathcal{H}^{n-m}(x)\,dy$ where $$\Omega \subset \mathbb{R}^n$$ is an open set, $$f:\Omega \rightarrow \mathbb{R}^m$$, $$J_mf(x)$$ is the $$m$$-dimensional Jacobian of $$f$$ and $$g: \Omega \rightarrow \mathbb{R}$$ is integrable and $$1\leq m<n$$, which is due to H. Federer in the case of Lipschitzian $$f$$, is extended to the case of Sobolev mappings $$f\in W_{\text{loc}}^{1,p}(\Omega; \mathbb{R}^m)$$, where $$p>m$$ or $$p\geq m=1$$. The limiting case $$p=m$$ for Hölder continuous mappings is also treated.

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26B10 Implicit function theorems, Jacobians, transformations with several variables 26B35 Special properties of functions of several variables, Hölder conditions, etc. 49Q15 Geometric measure and integration theory, integral and normal currents in optimization

### Keywords:

co-area formula; Sobolev mapping; rectifiability
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### References:

 [1] 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. · Zbl 0545.31011 [2] 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). [3] Bojarski, B., Haj\lasz, P., and Strzelecki, P., Pointwise inequalities for Sobolev functions revisited, Preprint 2000. [4] Cesari, L., Sulle trasformazioni continue, Ann. Mat. Pura Appl. 21 (1942), 157-188. · Zbl 0028.21004 [5] Eilenberg, S., On $$\varphi$$ measures, Ann. Soc. Pol. de Math. 17 (1938), 251-252. [6] Federer, H., Surface area (II), Trans. Amer. Math. Soc. 55 (1944), 438-456. · Zbl 0060.14003 [7] Federer, H., The $$(\phi,k)$$ rectifiable subsets of $$n$$space, Trans. Amer. Math. Soc. 62 (1947), 114-192. · Zbl 0032.14902 [8] Federer, H., Some integralgeometric theorems, Trans. Amer. Math. Soc. 77 (1954), 238-261. · Zbl 0058.16405 [9] Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418 – 491. · Zbl 0089.38402 [10] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 [11] Wendell H. Fleming, Functions whose partial derivatives are measures, Illinois J. Math. 4 (1960), 452 – 478. · Zbl 0151.05402 [12] Herbert Federer and William P. Ziemer, The Lebesgue set of a function whose distribution derivatives are \?-th power summable, Indiana Univ. Math. J. 22 (1972/73), 139 – 158. · Zbl 0238.28015 [13] 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. · Zbl 0962.49012 [14] Fiorenza, A., and Prignet, A., Orlicz capacities and applications to some existence questions for elliptic PDEs having measure data, preprint. · Zbl 1075.35012 [15] Haj\lasz, P., Sobolev mappings, co-area formula and related topics, Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk, 2000, 227-254. · Zbl 0988.28002 [16] Hencl, S. and Malý, J., Mapping of finite distortion: Hausdorff measure of zero sets, To appear in Math. Ann. [17] Janne Kauhanen, Pekka Koskela, and Jan Malý, On functions with derivatives in a Lorentz space, Manuscripta Math. 100 (1999), no. 1, 87 – 101. · Zbl 0976.26004 [18] K. Kuratowski, Topology. Vol. I, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. K. Kuratowski, Topology. Vol. II, New edition, revised and augmented. Translated from the French by A. Kirkor, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968. [19] 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. · Zbl 0819.28004 [20] Jan Malý, Hölder type quasicontinuity, Potential Anal. 2 (1993), no. 3, 249 – 254. · Zbl 0803.46037 [21] Jan Malý, The area formula for \?^{1,\?}-mappings, Comment. Math. Univ. Carolin. 35 (1994), no. 2, 291 – 298. · Zbl 0812.30006 [22] Jan Malý, Absolutely continuous functions of several variables, J. Math. Anal. Appl. 231 (1999), no. 2, 492 – 508. · Zbl 0924.26008 [23] Malý, J., Sufficient Conditions for Change of Variables in Integral, Proceedings on Analysis and Geometry, Sobolev Institute Press, Novosibirsk (2000) 370-386. · Zbl 0988.26011 [24] Malý, J., Wolff potential estimates of superminimizers of Orlicz type Dirichlet integrals, Preprint MATH-KMA-2002/74, Charles University, Praha, 2002. · Zbl 1098.35061 [25] Jan Malý and Olli Martio, Lusin’s condition (N) and mappings of the class \?^{1,\?}, J. Reine Angew. Math. 458 (1995), 19 – 36. · Zbl 0812.30007 [26] Malý, J., Swanson, D., and Ziemer, W. P., Fine behavior of functions with gradients in a Lorentz space, In preparation. [27] 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. · Zbl 0275.49041 [28] V. G. Maz$$^{\prime}$$ja and V. P. Havin, A nonlinear potential theory, Uspehi Mat. Nauk 27 (1972), no. 6, 67 – 138. [29] Ju. G. Rešetnjak, The concept of capacity in the theory of functions with generalized derivatives, Sibirsk. Mat. Ž. 10 (1969), 1109 – 1138 (Russian). [30] Rado, T. and Reichelderfer, P. V., Continuous Transformations in Analysis, Springer-Verlag, Berlin (1955). · Zbl 0067.03506 [31] Swanson, D., Pointwise inequalities and approximation in fractional Sobolev spaces, Studia Math. 149 (2002), 147-174. · Zbl 0993.46016 [32] 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). · Zbl 0762.46019 [33] William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. · Zbl 0692.46022
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