In this article, we introduce a model featuring a Lévy process in a bounded domain with semi-transparent boundary, by considering the fractional Laplacian operator with lower order non-local perturbations. We study the wellposedness of the model, certain qualitative properties and Runge type approximation. Furthermore, we consider the inverse problem of determining the unknown coefficients in our model from the exterior measurements of the corresponding Cauchy data. We also discuss the recovery of all the unknown coefficients from a single measurement.
We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet to Neumann map, only on a part of the boundary, corresponding to the isotropic elasticity equation of a 3-dimensional object. In earlier works it has been shown that one can determine the sheer and compressional speeds on a neighborhood of the part of the boundary (accessible part) where the measurements have been taken. In this article we show that one can determine the density of the medium as well, on a neighborhood of the accessible part of the boundary. *
We consider the following perturbed polyharmonic operator L(x, D) of order 2m defined in a bounded domain Ω ⊂ R n , n ≥ 3 with smooth boundary, aswhere A is a symmetric 2-tensor field, B and q are vector field and scalar potential respectively. We show that the coefficients A = [A jk ], B = (B j ) and q can be recovered from the associated Dirichletto-Neumann data on the boundary. Note that, this result shows an example of determining higher order (2nd order) symmetric tensor field in the class of inverse boundary value problem.2010 Mathematics Subject Classification. Primary 35R30, 31B20, 31B30, 35J40 .
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