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Here we study the Dirichlet problem for first order linear and quasi-linearhyperbolic PDEs on a simply connected bounded domain of $\R^2$, where thedomain has an interior outflow set and a mere inflow boundary. By means of aLyapunov function we show the existence of a unique solution in the space offunctions of bounded variation and its continuous dependence on all the data ofthe linear problem. Finally, we conclude the existence of a solution to thequasi-linear case by utilizing the Schauder fixed point theorem. This type ofproblems considered here appears in applications such as transport based imageinpainting.

On the Dirichlet Problem for First Order Linear Hyperbolic PDEs on Bounded Domains with Mere Inflow Boundary