Inverse Boundary Problems for Systems in Two Dimensions

Colin Guillarmou (1) , Leo Tzou (2) , Pierre Albin (3) , Gunther Uhlmann (4)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 Dept of Mathematics
3 IMJ - Institut de Mathématiques de Jussieu
4 University of Washington [Seattle]

We prove identification of coefficients up to gauge by Cauchy data at the boundary for a family of elliptic systems on oriented compact surfaces with boundary or domains of $\mathbb{C}$. In the geometric setting, we fix a Riemann surface with boundary, and consider both a Dirac-type operator plus potential acting on sections of a Clifford bundle and a connection Laplacian plus potential (i.e. Schrödinger Laplacian with external Yang-Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determines both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of $\mathbb{C}$, we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]

Colin Guillarmou, Leo Tzou, Pierre Albin, Gunther Uhlmann. Inverse Boundary Problems for Systems in Two Dimensions. Annales Henri Poincaré, 2013, 14 (6), pp.1551-1571. ⟨10.1007/s00023-012-0229-1⟩. ⟨hal-00595141⟩