### Mathematics in Engineering

2020, Issue 4: 584-597. doi: 10.3934/mine.2020026
Research article Special Issues

# On the obstacle problem for the 1D wave equation

• 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.

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

### Related Papers:

• 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.

 [1] Amerio L (1976) Su un problema di vincoli unilaterali per l'equazione non omogenea della corda vibrante. Pubbl IACD 190: 3-11. [2] Amerio L, Prouse G (1975) Study of the motion of a string vibrating against an obstacle. Rend Mat 8: 563-585. [3] Bamberger A, Schatzman M (1983) New results on the vibrating string with a continuous obstacle. SIAM J Math Anal 14: 560-595. doi: 10.1137/0514046 [4] Citrini C (1975) Sull'urto parzialmente elastico o anelastico di una corda vibrante contro un ostacolo. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur 59: 368-376. [5] Citrini C (1977) The energy theorem in the impact of a string vibrating against a point-shaped obstacle. Rend Act Naz Lincei 62: 143-149. [6] Kim J (1989) A boundary thin obstacle problem for a wave equation. Commun Part Diff Eq 14: 1011-1026. doi: 10.1080/03605308908820640 [7] Kim J (2008) On a stochastic wave equation with unilateral boundary conditions. T Am Math Soc 360: 575-607. doi: 10.1090/S0002-9947-07-04143-8 [8] Lebeau G, Schatzman M (1984) A wave problem in a half-space with a unilateral constraint at the boundary. J Differ Equations 53: 309-361. doi: 10.1016/0022-0396(84)90030-5 [9] Pinchover Y, Rubinstein J (2005) An Introduction to Partial Differential Equations, Cambridge: Cambridge University Press. [10] Schatzman M (1980) A hyperbolic problem of second order with unilateral constraints: The vibrating string with a concave obstacle. J Math Anal Appl 73: 138-191. doi: 10.1016/0022-247X(80)90026-8 [11] Schatzman M (1980) Un problème hyperbolique du 2ème ordre avec contrainte unilatérale: La corde vibrante avec obstacle ponctuel. J Differ Equations 36: 295-334. doi: 10.1016/0022-0396(80)90068-6
• © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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