Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets
Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets
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DOI：
10.1016/j.jde.2019.03.008
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发表时间：
2019-08
影响因子：
2.4
通讯作者：
Yavar Kian;Y. Kurylev;M. Lassas;L. Oksanen
中科院分区：
文献类型：
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作者：
Yavar Kian;Y. Kurylev;M. Lassas;L. Oksanen
考虑一个与算子Δ t2 − Δ g+ A+ q相关联的限制Dirichlet-to-Neumann映射Λ S，RT，其中Δ g是黎曼流形（M，g）上的Laplace-Beltrami算子，A和q是M上的向量场和函数.限制Λ S，RT对应于Dirichlet迹支持在（0，T）× S上而Neumann迹限制在（0，T）× R上的情况。这里S和R是开集，它们可以在M的边界上不相交。我们证明了在自然规范不变性的前提下，Λ S，RT唯一地决定了集合R的邻域中的低阶项A和q，假设R是严格凸的，并且波动方程在时间T/2上是S精确可控的.在凸叶理条件下，我们也给出了一个整体结果。主要的新奇是恢复的A和q时，集R和S是不相交的。我们允许A和q是非自伴的，特别地，相应的物理系统可能具有能量耗散。
We consider a restricted Dirichlet-to-Neumann map Λ S, R T associated with the operator∂ t 2− Δ g+ A+ q where Δ 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 Λ S, R T corresponds to the case where the Dirichlet traces are supported on (0, T)× S and the Neumann traces are restricted on (0, T)× R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that Λ 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.