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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 $\varphi$.

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

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order $\frac{1}{\sqrt{\epsilon }}$ concentrated on an $\epsilon$-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order $\frac{1}{\sqrt{\epsilon }}$ concentrated on an -neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.