Inverse Boundary Value Problem of Determining Up to a Second Order Tensor Appear in the Lower Order Perturbation of a Polyharmonic Operator

Abstract

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 .

An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator

“…A jk = a(x)δ jk for some scalar function a(x) in L(x, D) = (− ) m + a(x)(− ) + n j=1 B j (x)D j + q(x), m ≥ 2, (1.3) then a can be determined along with the B, q by knowing the boundary DN map. Then in [8] we extend this result by showing that it is possible to determine any symmetric matrix A from the perturbed polyharmonic operator:…”

“…In the earlier works on the Schrödinger and the magnetic Schrödinger operators [7,11,14,15,21,34,35] we encounter only first order transport equations. On the other hand, the works on biharmonic and polyharmonic operators [8,17,23,24] we get potential free higher order transport equations for the amplitudes that ((μ 1 + iμ 2 ) • ∇) 2 a 0 = 0, which can be dealt with in the same way as for the first order transport equations, see [8].…”

“…To this end we closely follow the arguments in [8]. Let us fix ξ ∈ R\{0} and consider the orthonormal basis B of R n as…”

Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data

“…Another advantage of the analysis of MRT in the context of our inverse problem is that we can relax the restriction that the domain be simply connected as was required in the works [20,21,13,2,3]. This assumption is required in these works since the recovery of a vector field perturbation (for instance) from boundary Dirichlet-to-Neumann data was done in two steps.…”

Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

“…Thanks to the work [33], logarithmic stability estimates are expected to be optimal for such inverse boundary value problems when the operator ∆ 2 is replaced by ∆ and k = 0, even when the full Cauchy data set C ∂Ω q (k) is given. We refer to the works [3], [4], [5], [9], [14], [19], [20], [27], [28], [29], [35], among others, for the study of inverse boundary value problems for perturbed biharmonic operators.…”