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The fractional anisotropic Calder\'{o}n problem for a nonlocal parabolic equation on closed Riemannian manifolds
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abstract
We consider the fractional anisotropic Calder\'on problem for the nonlocal parabolic equation $(\partial_t -\Delta_g)^s u=f$ ($0<s<1$) on closed Riemannian manifolds. More concretely, we can determine the Riemannian manifold $(M,g)$ up to isometry by using the local source-to-solution map in an arbitrarily small open cylinder in the spacetime domain. This can be regarded as a nonlocal analog of the anisotropic Calder\'on problem in the parabolic setting. We also study several useful properties for nonlocal parabolic operators by using comprehensive spectrum analysis with semigroup theory.
Forward citations
Cited by 2 Pith papers
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Entanglement principle for the fractional Laplacian with applications to inverse problems
A unique-continuation principle for sums of fractional Laplacians is proved on Euclidean space and applied to recover anisotropic coefficients and potentials in fractional polyharmonic equations.
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The Calder\'on problem for the logarithmic Schr\"odinger equation
For the logarithmic Schrödinger operator, the Dirichlet-to-Neumann map uniquely determines bounded potentials in arbitrary space dimension, and monotonicity gives a constructive reconstruction.
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