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wave equation; hysteresis; Hooke law; elasto-plastic materials; existence; uniqueness; weak omega-periodic solutions; Ishlinskij operator
We prove the existence and find necessary and sufficient conditions for the uniqueness of the time-periodic solution to the equations $u_{tt} - \Delta_xu \pm F(u) = g(x,t)$ for an arbitrary (sufficiently smooth) periodic right-hand side $g$, where $\Delta_x$ denotes the Laplace operator with respect to $x\in \Omega \subset R^N, N\geq 1$, and $F$ is the Ishlinskii hysteresis operator. For $N=2$ this equation describes e.g. the vibrations of an elastic membrane in an elastico-plastic medium.
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