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

Abstract

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

31 pages, 2 figures, submitted to SIAM J. Math. Anal

Cited by in corpus (1)

• On the Dirichlet Problem for First Order Hyperbolic PDEs on Bounded Domains with Mere Inflow Boundary: Part II Quasi-Linear Equations