Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Superposition operator in Sobolev spaces on domains


Author: Denis A. Labutin
Journal: Proc. Amer. Math. Soc. 128 (2000), 3399-3403
MSC (1991): Primary 46E35; Secondary 47H30
DOI: https://doi.org/10.1090/S0002-9939-00-05421-6
Published electronically: May 11, 2000
MathSciNet review: 1676320
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For an arbitrary open set $\Omega\subset \mathbb{R}^n$ we characterize all functions $G$ on the real line such that $G\circ u\in W^{1,p}(\Omega)$ for all $u\in W^{1,p}(\Omega)$. New element in the proof is based on Maz'ya's capacitary criterion for the imbedding $ {W^{1,p}(\Omega)\hookrightarrow L^\infty(\Omega)}$.


References [Enhancements On Off] (What's this?)

  • 1. J. Appel, P. P. Zabrejko, Nonlinear superposition operators, Cambridge Tracts in Mathematics, 95, Cambridge University Press, Cambridge, 1990. MR 91k:47168
  • 2. B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, 52-68, Lecture Notes in Math., 1351, Springer, Berlin-New York, 1988. MR 90b:46068
  • 3. G. Bourdaud, Le calcul fonctionnel dans les espaces de Sobolev, Invent. Math. 104 (1991), 435-446. MR 93b:46053
  • 4. G. Bourdaud, The functional calculus in Sobolev spaces, Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), 127-142, Teubner-Texte Math., 133, Teubner, Stuttgart, 1993. MR 94e:46055
  • 5. S. Buckley, P. Koskela, Sobolev-Poincare implies John, Math. Res. Lett. 2 (1995), no. 5, 577-593. MR 96i:46035
  • 6. V. M. Gol'dshtein, Yu. G. Reshetnyak, Quasiconformal mappings and Sobolev spaces, Kluwer Academic Publishers Group, Dordrecht, 1990. MR 92h:46040
  • 7. V. G. Maz'ya, Sobolev spaces, Springer-Verlag, Berlin-New York, 1985. MR 87g:46056
  • 8. M. Marcus, V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294-320. MR 49:3529
  • 9. M. Marcus, V. J. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217-229. MR 80h:47039
  • 10. M. Marcus, V. J. Mizel, Complete characterization of functions which act, via superposition, on Sobolev spaces, Trans. Amer. Math. Soc. 251 (1979), 187-218. MR 80j:46055
  • 11. T. Runst, W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, de Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter Co., Berlin, 1996. MR 98a:47071
  • 12. W. Sickel, Superposition of functions in Sobolev spaces of fractional order. A survey, Partial differential equations, 481-497, Banach Center Publ., 27, Part 1, 2, Polish Acad. Sci., Warsaw, 1992. MR 94e:46065
  • 13. W. Sickel, Composition operators acting on Sobolev spaces of fractional order--a survey on sufficient and necessary conditions, Function spaces, differential operators and nonlinear analysis (Paseky nad Jizerou, 1995), 159-182, Prometheus, Prague, 1996. CMP 98:04

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 46E35, 47H30

Retrieve articles in all journals with MSC (1991): 46E35, 47H30


Additional Information

Denis A. Labutin
Affiliation: Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra 0200, ACT, Australia
Email: labutin@maths.anu.edu.au

DOI: https://doi.org/10.1090/S0002-9939-00-05421-6
Keywords: Sobolev spaces, superposition operator
Received by editor(s): August 1, 1998
Received by editor(s) in revised form: January 22, 1999
Published electronically: May 11, 2000
Additional Notes: This work was supported by the Russian Foundation for Basic Research grant 96-01-00243.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society