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Harmonic homeomorphisms of the closed disc to itself need be in $ W^{1,p}$, $ p<2$, but not $ W^{1,2}$

Author(s): Gregory C. Verchota
Journal: Proc. Amer. Math. Soc. 135 (2007), 891-894.
MSC (2000): Primary 58E20, 46E35
Posted: August 28, 2006
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Abstract: Harmonic maps $ u$ from the closed disc onto bounded convex sets of the plane obey $ u\in W^{1,p}, p<2$.

References:

[Ada75]
Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957 (56:9247)

[AIMO]
K. Astala, T. Iwaniec, G. J. Martin, and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. 91 (2005), no. 3, 655-702. MR 2180459

[BB98]
Haïm Brezis and Felix Browder, Partial differential equations in the 20th century, Adv. Math. 135 (1998), no. 1, 76-144. MR 1617413 (99i:35001)

[Cou50]
R. Courant, Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces, Interscience Publishers, Inc., New York, N.Y., 1950. MR 0036317 (12:90a)

[Die81]
Jean Dieudonné, History of functional analysis, North-Holland Mathematics Studies, vol. 49, North-Holland Publishing Co., Amsterdam, 1981. MR 0605488 (83d:46001)

[Dur04]
Peter Duren, Harmonic mappings in the plane, Cambridge University Press, Cambridge, 2004. MR 2048384 (2005d:31001)

[EG92]
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 (93f:28001)

[Had06]
J. Hadamard, Sur le principe de Dirichlet, Bull. Soc. Math. France 34 (1906), 135-138. MR 1504545

[HLP52]
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. MR 0046395 (13:727e)

[Joh82]
Fritz John, Partial differential equations, Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR MR831655 (87g:35002)

[Kne26]
Hellmuth Kneser, Lösung der Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 123-124.

[MP97]
Vladimir G. Maz'ya and Sergei V. Poborchi, Differentiable functions on bad domains, World Scientific Publishing Co. Inc., River Edge, NJ, 1997. MR 1643072 (99k:46057)

[Pee76]
Jaak Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. MR 0461123 (57:1108)

[Pry71]
Friedrich Prym, Zur Integration der Differentialgleichung $ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$, J. Reine Angew. Math. 73 (1871), 340-364.

[Ste70]
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095 (44:7280)

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Additional Information:

Gregory C. Verchota
Affiliation: Department of Mathematics, 215 Carnegie, Syracuse University, Syracuse, New York 13244
Email: gverchot@syr.edu

DOI: 10.1090/S0002-9939-06-08506-6
PII: S 0002-9939(06)08506-6
Received by editor(s): June 6, 2005
Received by editor(s) in revised form: September 26, 2005
Posted: August 28, 2006
Additional Notes: The author thanks Tadeusz Iwaniec for this question.
Communicated by: Juha M. Heinonen
Copyright of article: Copyright 2006, American Mathematical Society

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