Calculus of Variations and Geometric Measure Theory

X. Fernández-Real - A. Figalli

On the obstacle problem for the 1D wave equation

created by figalli on 24 Feb 2020
modified on 10 Feb 2023


Published Paper

Inserted: 24 feb 2020
Last Updated: 10 feb 2023

Journal: Mathematics in Engineering
Year: 2020


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.