Uniqueness and boundary behaviour of solutions to variational problems with linear growth

Abstract

We investigate the Dirichlet problem for the variational integral $J[u] = \int_{\Omega} f(\nabla u) \, dx$ with density $f$ of linear growth satisfying appropriate ellipticity conditions. We show that the relaxed problem admits a unique solution $u$ in the space of functions of bounded variation, if the set $\Gamma_0$ of convex points $x \in \partial\Omega$ is sufficiently large. For example, the inequality $\mathcal{H}^{n-1}(\Gamma_0) > \frac{2}{3}\mathcal{H}^{n-1}(\partial\Omega)$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\Omega$ and attains the prescribed boundary data at least on $\Gamma_0$ in the classical sense.