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

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 $|D\varphi (x)|^n\le L\det D\varphi (x)$ for almost every $x\in \Omega$ there exists a map $u\in W^{1,n}(\Omega ;\mathbf {R}^n)$ such that \begin{equation*} \begin {cases} |Du(x)|^n=L\det Du(x)\quad \text {a.e.}\ x\in \Omega , u|_{\partial \Omega }=\varphi ,\quad \|u-\varphi \|_{L^n(\Omega )}<\epsilon . \end{cases} \end{equation*} 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.