### A differential inclusion : the case of an isotropic set

In this article we are interested in the following problem: to find a map $u:\Omega \to {\mathbb{R}}^{2}$ that satisfies$$\left\{\begin{array}{cc}Du\in E\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\hfill & \mathit{\text{a.e.}}\phantom{\rule{4.0pt}{0ex}}\text{in}\phantom{\rule{4.0pt}{0ex}}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \phantom{\rule{85.35826pt}{0ex}}\hfill \end{array}\right.$$where $\Omega $ is an open set of ${\mathbb{R}}^{2}$ and $E$ is a compact isotropic set of ${\mathbb{R}}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi $.