On the obstacle problem for the 1D wave equation

TL;DR

This paper reviews the theory of the one-dimensional obstacle problem for the wave equation, proving Lipschitz norm preservation after collision and discussing explicit solution formulas.

Contribution

It introduces a framework for existence and uniqueness, proves Lipschitz norm preservation, and analyzes explicit solutions for the obstacle problem.

Findings

Lipschitz norm is preserved after collision in the obstacle problem.

Solutions have bounded Lipschitz norm at all times.

Discussion on the validity of explicit solution formulas.

Abstract

Our goal is to review the known theory on the one-dimensional obstacle problem for the wave equation, and to discuss some extensions. We introduce the setting established by Schatzman within which existence and uniqueness of solutions can be proved, and we prove that (in some suitable systems of coordinates) the Lipschitz norm is preserved after collision. As a consequence, we deduce that solutions to the obstacle problem (both simple and double) for the wave equation have bounded Lipschitz norm at all times. Finally, we discuss the validity of an explicit formula for the solution that was found by Bamberger and Schatzman.