Inverse boundary value problems for the perturbed polyharmonic operator (Q2862125)

Let \(\Omega\subset \mathbb R^n, n\geq 3\), be a bounded domain with \(C^\infty\) boundary. Consider the perturbed polyharmonic operator NEWLINE\[NEWLINE \mathcal L_{A,q}(x,D) = (-\Delta)^m + A(x)\cdot D + q(x),NEWLINE\]NEWLINE where \(m\geq 1\), \(A = (A_j)_{1\leq j \leq n} \in W^{1,\infty}(\Omega,\mathbb C^n)\), \(q(n)\in L^\infty(\Omega, \mathbb C) \) and \(D = i^{-1} \nabla\). This paper considers the inverse problem of recovering the perturbation \(A(x)\cdot D + q(x)\) of \(\mathcal L_{A,q}(x,D)\) from the Dirichlet-to-Neumann map. Under some smoothness conditions on \(A(x)\) and \(q(x)\), the authors prove that both \(A(x)\) and \(q(x)\) are uniquely determined from the Dirichlet-to-Neumann map for \(m \geq 2\). The proofs are based on the construction of complex geometric optics solutions.