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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Deformations of finite conformal energy: Boundary behavior and limit theorems


Authors: Tadeusz Iwaniec and Jani Onninen
Journal: Trans. Amer. Math. Soc. 363 (2011), 5605-5648
MSC (2000): Primary 35J15, 35J70
Published electronically: June 15, 2011
MathSciNet review: 2817402
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Abstract: We study homeomorphisms $ h: \mathbb{X} \stackrel{\textrm{\tiny {onto}}}{\longrightarrow} \mathbb{Y} $ between two bounded domains in $ \mathbb{R}^n$ having finite conformal energy

$\displaystyle \mathcal E [h]= \int_{\mathbb{X}} \vert\vert Dh(x) \vert\vert ^n ... ...al{d}x < \infty , \hskip0.3cm h \in \mathscr W^{1,n}(\mathbb{X} , \mathbb{Y}).$

We consider the behavior of such mappings, including continuous extension to the closure of $ \mathbb{X}$ and injectivity of $ h: \overline{\mathbb{X}} \to \overline{\mathbb{Y}}$. In general, passing to the weak $ \mathscr W^{1,n}$-limit of a sequence of homeomorphisms $ h_j: \mathbb{X} \to \mathbb{Y}$ one loses injectivity. However, if the mappings in question have uniformly bounded $ \mathscr L^1$-average of the inner distortion, then, for sufficiently regular domains $ \mathbb{X}$ and $ \mathbb{Y}$, their limit map $ h: \mathbb{X} \stackrel{\textrm{\tiny {onto}}}{\longrightarrow} \mathbb{Y}$ is a homeomorphism. Moreover, the inverse map $ f=h^{-1}: \mathbb{Y} \stackrel{\textrm{\tiny {onto}}}{\longrightarrow} \mathbb{X}$ enjoys finite conformal energy and has integrable inner distortion as well.


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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
Email: jkonnine@syr.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05106-8
PII: S 0002-9947(2011)05106-8
Keywords: Boundary behavior of homeomorphisms, limit theorems, energy integrals, quasiconformal hyperelasticity.
Received by editor(s): August 4, 2008
Received by editor(s) in revised form: May 13, 2009
Published electronically: June 15, 2011
Additional Notes: The first author was supported by NSF grant DMS-0800416.
The second author was supported by NSF grant DMS-0701059.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.