A differential inclusion : the case of an isotropic set

Croce, Gisella

ESAIM: Control, Optimisation and Calculus of Variations, Tome 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

$$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$

where $\Omega$ is an open set of ${ℝ}^2$ and $E$ is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

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