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A Baire's category method for the Dirichlet problem of quasiregular mappings

Author: Baisheng Yan
Journal: Trans. Amer. Math. Soc. 355 (2003), 4755-4765
MSC (2000): Primary 30C65, 35F30, 49J30
Published electronically: July 24, 2003
MathSciNet review: 1997582
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Abstract: We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any $\epsilon>0$ and any piece-wise affine map $\varphi\in W^{1,n}(\Omega;\mathbf{R}^n)$ with $\vert D\varphi(x)\vert^n\le L\det D\varphi(x)$ for almost every $x\in\Omega$ there exists a map $u\in W^{1,n}(\Omega;\mathbf{R}^n)$ such that

\begin{displaymath}\begin{cases} \vert Du(x)\vert^n=L\det Du(x)\quad\text{a.e.} ... ...,\quad\Vert u-\varphi\Vert _{L^n(\Omega)}<\epsilon. \end{cases}\end{displaymath}

The theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.

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Additional Information

Baisheng Yan
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Keywords: Baire's category method, Dirichlet problem, quasiregular mappings
Received by editor(s): January 25, 2001
Published electronically: July 24, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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