Hyperbolic inverse problem with data on disjoint sets

Abstract : We consider a restricted Dirichlet-to-Neumann map $\Lambda_{S, \R}^T$ associated with the operator $\partial_t^2 - \Delta_g + A + q$ where $\Delta_g$ is the Laplace-Beltrami operator of a Riemannian manifold $(M,g)$, and $A$ and $q$ are a vector field and a function on $M$. The restriction $\Lambda_{S, R}^T$ corresponds to the case where the Dirichlet traces are supported on $(0, T) \times S$ and the Neumann traces are restricted on $(0, T) \times R$. Here $S$ and $R$ are open sets, which may be disjoint, on the boundary of $M$. We show that $\Lambda_{S, R}^{T}$ determines uniquely, up the natural gauge invariance, the lower order terms $A$ and $q$ in a neighborhood of the set $R$ assuming that $R$ is strictly convex and that the wave equation is exactly controllable from $S$ in time $T/2$. We give also a global result under a convex foliation condition. The main novelty is the recovery of $A$ and $q$ when the sets $R$ and $S$ are disjoint. We allow $A$ and $q$ to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.
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Yavar Kian, Yaroslav Kurylev, Matti Lassas, Lauri Oksanen. Hyperbolic inverse problem with data on disjoint sets. 2018. ⟨hal-01784652⟩

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