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

TL;DR

This paper establishes the uniqueness and continuous dependence of solutions for first order hyperbolic quasi-linear PDEs with causal functional dependence on bounded domains with inflow boundaries, relevant for applications like transport-based image inpainting.

Contribution

It introduces a contraction principle for causal functional dependence, proving uniqueness and stability of solutions in a quasi-linear hyperbolic PDE setting.

Findings

Proved uniqueness of solutions under causal dependence.

Established continuous dependence on PDE coefficients.

Applied results to transport-based image inpainting.

Abstract

We study the Dirichlet problem for first order hyperbolic quasi-linear functional PDEs on a simply connected bounded domain of $\R^2$, where the domain has an interior outflow set and a mere inflow boundary. While the question of existence of a solution has already been answered in its predecessor, the present paper discusses the uniqueness and continuous dependence on the coefficients of the PDE. Under the assumption that the functional dependence is causal, we are able to derive a contraction principle which is the key to proof uniqueness and continuous dependence. Such a causal functional dependence appears, e.g., in transport based image inpainting.