Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

By Matti Lassas, Lauri Oksanen. 2012

77 citations recorded by the source corpus.

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M, g)$ from a restriction $\Lambda_{\Src, \Rec}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\Src$ and $\Rec$ are open sets in $\p M$ and the restriction $\Lambda_{\Src, \Rec}$ corresponds to the case where the Dirichlet data is supported on $\R_+\times \Src$ and the Neumann data is measured on $\R_+\times \Rec$. In the novel case where $\bar \Src \cap \bar \Rec = \emptyset$, we show that $\Lambda_{\Src, \Rec}$ determines the manifold $(M,g)$ uniquely, assuming that the wave equation is exactly controllable from the set of sources $\Src$. Moreover, we show that the exact controllability can be replaced by the Hassell-Tao condition for eigenvalues and eigenfunctions, that is, \lambda_j \le C \norm{\p_\nu \phi_j}_{L^2(\Src)}^2, \quad j =1, 2, ..., where $\lambda_j$ are the Dirichlet eigenvalues and $(\phi_j)_{j=1}^\infty$ is an orthonormal basis of the corresponding eigenfunctions.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/ad764b9888c210d4409e8bdf8eb0fe581e758270

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In Computational Theory and Mathematics

In Mathematical Physics

In Mechanics of Materials

Citations in the corpus

Inverse problems for the connection Laplacian (2015, methodology): “3 are weakened, then all the known uniqueness results in the scalar case require additional assumptions on the global geometry of (M, g), see [14, 25, 28].”
Global uniqueness in an inverse problem for time fractional diffusion equations (2016, methodology): “3) implies that LM1,g1 = LM2,g2 , and therefore (Mk, gk), k = 1, 2, are isometric [39].”