Deformations of finite conformal energy: Boundary behavior and limit theorems

Iwaniec, Tadeusz; Onninen, Jani

doi:10.1090/S0002-9947-2011-05106-8

Deformations of finite conformal energy: Boundary behavior and limit theorems
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Trans. Amer. Math. Soc. 363 (2011), 5605-5648 Request permission

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 \[ \mathcal E [h]= \int _{\mathbb X} |\!| Dh(x) |\!| ^n \mathrm {d}x < \infty , \hskip 0.3cm h \in \mathscr W^{1,n}(\mathbb X , \mathbb Y).\] 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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