A differential inclusion : the case of an isotropic set
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 1, pp. 122-138.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^{2}$ 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{4pt}{0ex}}\text{in}\phantom{\rule{4pt}{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 ${ℝ}^{2}$ and $E$ is a compact isotropic set of ${ℝ}^{2×2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

DOI: 10.1051/cocv:2004035
Classification: 34A60,  35F30,  52A30
Keywords: rank one convex hull, polyconvex hull, differential inclusion, isotropic set
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Croce, Gisella. A differential inclusion : the case of an isotropic set. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 1, pp. 122-138. doi : 10.1051/cocv:2004035. http://www.numdam.org/articles/10.1051/cocv:2004035/

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