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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Radó-Kneser-Choquet Theorem for simply connected domains ($ p$-harmonic setting)


Authors: Tadeusz Iwaniec and Jani Onninen
Journal: Trans. Amer. Math. Soc. 371 (2019), 2307-2341
MSC (2010): Primary 30E10; Secondary 46E35, 58E20
DOI: https://doi.org/10.1090/tran/7348
Published electronically: November 27, 2018
MathSciNet review: 3896082
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Abstract: A remarkable result known as the Radó-Kneser-Choquet theorem asserts that the harmonic extension of a homeomorphism of the boundary of a Jordan domain $ \,\Omega \subset \mathbb{R}^2\,$ onto the boundary of a convex domain $ \,\mathcal Q\subset \mathbb{R}^2\,$ takes $ \,\Omega \,$ diffeomorphically onto $ \,\mathcal Q\,$. Numerous extensions of this result for linear and nonlinear elliptic PDEs are known, but only when $ \,\Omega \,$ is a Jordan domain or, if not, under additional assumptions on the boundary map. On the other hand, the newly developed theory of Sobolev mappings between Euclidean domains and Riemannian manifolds demands extending this theorem to the setting of simply connected domains. This is the primary goal of our article. The class of the $ \,p\,$-harmonic equations is wide enough to satisfy those demands. Thus we confine ourselves to considering the $ \,p\,$-harmonic mappings.

The situation is quite different from that of Jordan domains. One must circumvent the inherent topological difficulties arising near the boundary.

Our main theorem is the key to establishing approximation of monotone Sobolev mappings with diffeomorphisms. This, in turn, leads to the existence of energy-minimal deformations in the theory of nonlinear elasticity.


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Additional Information

Tadeusz Iwaniec
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Email: tiwaniec@syr.edu

Jani Onninen
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244—and—Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
Email: jkonnine@syr.edu

DOI: https://doi.org/10.1090/tran/7348
Keywords: Harmonic mappings, $p$-harmonic equation, monotone mappings
Received by editor(s): March 5, 2017
Received by editor(s) in revised form: May 21, 2017, and July 1, 2017
Published electronically: November 27, 2018
Additional Notes: The first author was supported by NSF grant DMS-1301558.
The second author was supported by NSF grant DMS-1700274.
Article copyright: © Copyright 2018 American Mathematical Society