Whitney property in two dimensional Sobolev spaces
HTML articles powered by AMS MathViewer
- by Dorin Bucur, Alessandro Giacomini and Paola Trebeschi
- Proc. Amer. Math. Soc. 136 (2008), 2535-2545
- DOI: https://doi.org/10.1090/S0002-9939-08-09366-0
- Published electronically: March 4, 2008
- PDF | Request permission
Abstract:
For $p >1$, we prove that all the functions of $W_\textrm {loc}^{2,p}(\mathbb {R}^2)$ satisfy the Whitney property; i.e., if $u \in W_\textrm {loc}^{2,p}(\mathbb {R}^2)$ is such that $\nabla u=0$ (in the sense of capacity) on a connected set $K\subseteq \mathbb {R}^2$, then $u$ is constant on $K$.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
- Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
- Thomas Bagby, Quasi topologies and rational approximation, J. Functional Analysis 10 (1972), 259–268. MR 0355058, DOI 10.1016/0022-1236(72)90025-0
- S. M. Bates, On the image size of singular maps. I, Proc. Amer. Math. Soc. 114 (1992), no. 3, 699–705. MR 1074748, DOI 10.1090/S0002-9939-1992-1074748-8
- S. M. Bates, On the image size of singular maps. II, Duke Math. J. 68 (1992), no. 3, 463–476. MR 1194950, DOI 10.1215/S0012-7094-92-06818-9
- Bogdan Bojarski, Piotr Hajłasz, and PawełStrzelecki, Sard’s theorem for mappings in Hölder and Sobolev spaces, Manuscripta Math. 118 (2005), no. 3, 383–397. MR 2183045, DOI 10.1007/s00229-005-0590-1
- Friedemann Brock, Continuous rearrangement and symmetry of solutions of elliptic problems, Proc. Indian Acad. Sci. Math. Sci. 110 (2000), no. 2, 157–204. MR 1758811, DOI 10.1007/BF02829490
- D. Bucur, Shape analysis for nonsmooth elliptic operators, Appl. Math. Lett. 9 (1996), no. 3, 11–16. MR 1385991, DOI 10.1016/0893-9659(96)00023-7
- Dorin Bucur and Paola Trebeschi, Shape optimisation problems governed by nonlinear state equations, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), no. 5, 945–963. MR 1642112, DOI 10.1017/S0308210500030006
- Gianni dal Maso, François Ebobisse, and Marcello Ponsiglione, A stability result for nonlinear Neumann problems under boundary variations, J. Math. Pures Appl. (9) 82 (2003), no. 5, 503–532 (English, with English and French summaries). MR 1995490, DOI 10.1016/S0021-7824(03)00014-X
- Luigi de Pascale, The Morse-Sard theorem in Sobolev spaces, Indiana Univ. Math. J. 50 (2001), no. 3, 1371–1386. MR 1871360, DOI 10.1512/iumj.2001.50.1878
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- Alessandro Giacomini and Paola Trebeschi, A density result for Sobolev spaces in dimension two, and applications to stability of nonlinear Neumann problems, J. Differential Equations 237 (2007), no. 1, 27–60. MR 2327726, DOI 10.1016/j.jde.2007.02.011
- J. Heinonen, T. Kilpeläinen, and O. Martio, Fine topology and quasilinear elliptic equations, Ann. Inst. Fourier (Grenoble) 39 (1989), no. 2, 293–318 (English, with French summary). MR 1017281
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- Tero Kilpeläinen and Jan Malý, Supersolutions to degenerate elliptic equation on quasi open sets, Comm. Partial Differential Equations 17 (1992), no. 3-4, 371–405. MR 1163430, DOI 10.1080/03605309208820847
- John L. Lewis, Uniformly fat sets, Trans. Amer. Math. Soc. 308 (1988), no. 1, 177–196. MR 946438, DOI 10.1090/S0002-9947-1988-0946438-4
- Alec Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), no. 1, 367–376. MR 837817, DOI 10.1090/S0002-9947-1986-0837817-2
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
- Arthur Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890. MR 7523, DOI 10.1090/S0002-9904-1942-07811-6
- V. Šverák, On optimal shape design, J. Math. Pures Appl. (9) 72 (1993), no. 6, 537–551. MR 1249408
- Hassler Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), no. 4, 514–517. MR 1545896, DOI 10.1215/S0012-7094-35-00138-7
Bibliographic Information
- Dorin Bucur
- Affiliation: Laboratoire de Mathématiques, CNRS UMR 5127 Université de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France
- MR Author ID: 349634
- Email: dorin.bucur@univ-savoie.fr
- Alessandro Giacomini
- Affiliation: Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy
- Email: alessandro.giacomini@ing.unibs.it
- Paola Trebeschi
- Affiliation: Dipartimento di Matematica, Facoltà di Ingegneria, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy
- Email: paola.trebeschi@ing.unibs.it
- Received by editor(s): May 15, 2007
- Published electronically: March 4, 2008
- Communicated by: Mario Bonk
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2535-2545
- MSC (2000): Primary 46E35
- DOI: https://doi.org/10.1090/S0002-9939-08-09366-0
- MathSciNet review: 2390524