The paper investigates the vectorial Dirichlet problem defined by $$\begin{cases}\sigma_j(\nabla u(x))=1, \backslash, x\in\Omega \text{a.e.},\, j=1,\ldots, n u(x)=\varphi(x),\,x\in\partial\Omega. \end{cases}$$ Here $\Omega$ is an open bounded subset of $\mathbb{R}^n$ with boundary $\partial\Omega$, and $\sigma_j(A)$ ($j=1, \ldots , n$) denote the singular values of the gradient $\nabla u(x)$. The existence of solutions is established under one of the following assumptions: $\varphi: \overline{\Omega} \longrightarrow \mathbb{R}^n$ is continuous on $\overline{\Omega}$ and locally contractive on $\Omega$, or $\varphi: \partial\Omega \longrightarrow \mathbb{R}^n$ is contractive on $\partial\Omega$. This extends a result due to Dacorogna and Marcellini. The approach is based on the Baire category method developed earlier by the authors.