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

Abstract

We investigate the Dirichlet problem for the variational integral $J[u]={\int }_{\hat{I}}\text{c◯}f(\nabla u)dx$ 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 $\hat{I}{“}_0$ of convex points $x\in \partial \hat{I}\text{c◯}$ is sufficiently large. For example, the inequality ${\mathcal{H}}^{n-1}(\hat{I}{“}_0)>\frac{2}{3}{\mathcal{H}}^{n-1}(\partial \hat{I}\text{c◯})$ is sufficient. Moreover, the minimizer $u$ is smooth in the interior of $\hat{I}\text{c◯}$ and attains the prescribed boundary data at least on $\hat{I}{“}_0$ in the classical sense.