Abstract. In this paper we show existence of a trace for functions of bounded variation on Riemannian manifolds with boundary. The trace, which is bounded in L ∞ , is reached via L 1 -convergence and allows an integration by parts formula. We apply these results in order to show well-posedness and total variation estimates for the initial boundary value problem for a scalar conservation law on compact Riemannian manifolds with boundary in the context of functions of bounded variation via the vanishing viscosity method. The flux function is assumed to be time-dependent and divergence-free.

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