Existence of functions of fixed trace and minimum weighted total variation

For \Omega a C^{2}-smooth domain, and a positive bounded continuous map a \in C(\Omega), we prove existence of a minimizer of the functional u \mapsto $\int_{\Omega} a|Du| over the space BV(\Omega) of functions of bounded variation with fixed trace f \in L^{1}(\partial\Omega). The method of proof is constructive.