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.
Highlights
Consider an infinite vibrating string represented by its transversal displacement, u(x, t) ∈ R, with x ∈ R and t ∈ [0, ∞), with initial conditions given by u(x, 0) = u0(x) for x ∈ R ut(x, 0) = u1(x) for x ∈ R
Let us formally show that the formula (2.6) solves the obstacle problem for the wave equation, in the sense (1.1)-(1.2), for φ ≡ 0
In [BS83], Bamberger and Schatzman established an explicit formula for the solution u to the obstacle problem in terms of the free wave solution w
Summary
Let us assume that the obstacle is given by a wall, so that we can take φ ≡ 0 This problem was first studied by Amerio and Prouse in [AP75] in the finite string case (with fixed end-points), constructing a solution “by hand” by following the characteristic curves and extending the initial condition through the lines of influence. If one wants to impose reflections perpendicular to the obstacle ( hoping to avoid accumulation of collisions), one would need to consider a rotation invariant equation (in the graph space (x, u(x, t)) ∈ R2) instead of the wave equation (compare for instance [PR05, Equation (1.28) vs (1.29)]) Investigating these modeling questions is a very interesting problem, and we hope that this paper will be a starting point to motivate this beautiful line of research. The results presented here can be extended to the case of a finite string with fixed end-points, thanks to the locality of our methods
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