The Euler equation for functionals with linear growth

Abstract:

We give a first variation formula for functionals of the type $\int _\Omega {f(x,\mu )}$, where $f(x,p):\Omega \times {{\mathbf {R}}^k} \to {\mathbf {R}}$ is of linear growth in $p$ for large $|p|$ and $\mu$ is a ${{\mathbf {R}}^k}$-valued measure in $\Omega$. The Euler equation for the minima of various functionals defined on spaces of ${\text {BV}}$ functions is then studied.