On the obstacle problem for the 1D wave equation

Xavier Fernández-Real; Alessio Figalli; Xavier Fernández-Real; Alessio Figalli

doi:10.3934/mine.2020026

Mathematics in Engineering

2020, Volume 2, Issue 4: 584-597. doi: 10.3934/mine.2020026

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On the obstacle problem for the 1D wave equation

Xavier Fernández-Real ,
Alessio Figalli ^,

Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

Received: 01 November 2019 Accepted: 17 February 2020 Published: 19 May 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.
- obstacle problem,
- wave equation,
- one-dimensional
Citation: Xavier Fernández-Real, Alessio Figalli. On the obstacle problem for the 1D wave equation[J]. Mathematics in Engineering, 2020, 2(4): 584-597. doi: 10.3934/mine.2020026

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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.

References

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