Inverse problems for polyharmonic operators with lower order perturbations (Joint with IP seminar)  

 Abstract:  We consider an inverse problem of recovering lower order perturbations of a polyharmonic operator on a bounded domain \$\Omega\$ of dimension 3 or higher, from the Dirichlet to Neumann map defined on the boundary. We consider the operator
$$
\mathcal{L} = (-\Delta)^m + \sum_{j,k=1}^{n}A_{jk}(x) \partial_{x_j} \partial_{x_k} + \sum_{j=1}^{n} B_{j}(x) \partial_{x_j} + q(x) , \quad m \geq 2,
$$
where \$A:= (A_{jk})\$ is a symmetric matrix, \$B:=\{B_1,\dots,B_n\}\$ is a vector field and \$q\$ is a bounded function on the domain \$\Omega\$. We discuss the case of \$m>2\$, where we can recover the perturbations \$A\$, \$B\$ and \$q\$ in \$\Omega\$ from the Dirichlet to Neumann map defined on the boundary \$\partial \Omega\$; as well as the case of \$m=2\$, where there is an obstruction in the unique recovery of the perturbations from the boundary data.