6 research outputs found
Inverse problem for wave equation with sources and observations on disjoint sets
We consider an inverse problem for a hyperbolic partial differential equation
on a compact Riemannian manifold. Assuming that and are
two disjoint open subsets of the boundary of the manifold we define the
restricted Dirichlet-to-Neumann operator . This
operator corresponds the boundary measurements when we have smooth sources
supported on and the fields produced by these sources are observed
on . We show that when and are disjoint but
their closures intersect at least at one point, then the restricted
Dirichlet-to-Neumann operator determines the
Riemannian manifold and the metric on it up to an isometry. In the Euclidian
space, the result yields that an anisotropic wave speed inside a compact body
is determined, up to a natural coordinate transformations, by measurements on
the boundary of the body even when wave sources are kept away from receivers.
Moreover, we show that if we have three arbitrary non-empty open subsets
, and of the boundary, then the restricted
Dirichlet-to-Neumann operators for determine the Riemannian manifold to an isometry. Similar result is proven
also for the finite-time boundary measurements when the hyperbolic equation
satisfies an exact controllability condition
Multidimensional Borg-Levinson Theorem
We consider the inverse problem of the reconstruction of a Schr\"odinger
operator on a unknown Riemannian manifold or a domain of Euclidean space. The
data used is a part of the boundary and the eigenvalues corresponding
to a set of impedances in the Robin boundary condition which vary on .
The proof is based on the analysis of the behaviour of the eigenfunctions on
the boundary as well as in perturbation theory of eigenvalues. This reduces the
problem to an inverse boundary spectral problem solved by the boundary control
method