Inverse boundary problems for polyharmonic operators with unbounded potentials
Abstract
We show that the knowledge of the DirichlettoNeumann map on the boundary of a bounded open set in $R^n$ for the perturbed polyharmonic operator $(\Delta)^m +q$ with $q\in L^{n/2m}$, $n>2m$, determines the potential $q$ in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted $L^2$ and $L^p$ spaces. The $L^p$ estimates for the special Green function are derived from $L^p$ Carleman estimates with linear weights for the polyharmonic operator.
 Publication:

arXiv eprints
 Pub Date:
 August 2013
 DOI:
 10.48550/arXiv.1308.3782
 arXiv:
 arXiv:1308.3782
 Bibcode:
 2013arXiv1308.3782K
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematical Physics;
 35R30;
 35J40;
 31B30