Abstract
We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for 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 module 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 determine 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.
Article PDF
Similar content being viewed by others
References
Astala K., Päivärinta L.: Calderón’s inverse conductivity problem in the plane. Ann. Math. (2) 163(1), 265–299 (2006)
Astala K., Lassas M., Päivärinta L.: Calderón’s inverse problem for anisotropic conductivity in the plane. Comm. Partial Differ Equ 30(1–3), 207–224 (2005)
Beals R., Coifman R.: The Spectral Problem for Davey-Stewarson and The Ishimory Hierarchies . Nonlinear Evolution Equations: Integrability and Spectral Methods, pp. 15–23. Manchester University Press, Manchester (1988)
Belishev M.I.: The Calderón problem for two dimensional manifolds by the BC- method. SIAM J. Math. Anal. 35(1), 172–182 (2003)
Brown R., Uhlmann G.: Uniqueness in the inverse conductivity problem with less regular conductivities. Comm. PDE 22, 1009–1027 (1997)
Bukhgeim A.L.: Recovering a potential from Cauchy data in the two-dimensional case. J. Inverse Ill-Posed Probl. 16(1), 19–33 (2008)
Forster, O.: Lectures on Riemann Surfaces. GTM 81, Springer, Berlin (1981)
Guillarmou, C., Tzou, L.: Calderón inverse problem for Schrödinger operator on Riemann surfaces. Proceedings of the Centre for Mathematics and its Applications, vol. 44—Proceedings of the AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis (2010)
Guillarmou C., Tzou L.: Calderón inverse Problem with partial data on Riemann Surfaces. Duke Math. J. 158(1), 83–120 (2011)
Guillarmou C., Tzou L.: Identification of a connection from Cauchy data space on a Riemann surface with boundary. GAFA 21(2), 393–418 (2011)
Henkin G., Michel V.: On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator. GAFA 17, 116–155 (2007)
Henkin G., Michel V.: Inverse conductivity problem on Riemann surfaces. J. Geom. Anal. 18(4), 1033–1052 (2008)
Henkin G., Novikov R.G.: On the reconstruction of conductivity of bordered two-dimensional surface in \({\mathbb{R}^3}\) from electrical currents measurements on its boundary. J. Geom. Anal. 21(3), 543–587 (2011)
Henkin G., Santacesaria M.: Gel’fand-Calderón’s inverse problem for anisotropic conductivities on bordered surfaces in \({\mathbb{R}^3}\) . Inverse Probl. 26(9), 095011–095018 (2010)
Hill C.D., Taylor M.: Integrability of rough almost complex structures. J. Geom. Anal. 13(1), 163–172 (2003)
Imanuvilov O.Y., Uhlmann G., Yamamoto M.: Global uniqueness from partial Cauchy data in two dimensions. J. Am. Math. Soc. 23, 655–691 (2010)
Imanuvilov, O., Uhlmann, G., Yamamoto, M.: Partial Cauchy data for general second order operators in two dimensions. To appear in Publ. Research Insti. Math. Sci.
Imanuvilov O., Uhlmann G., Yamamoto M.: On determination of second order operators from partial Cauchy data. Proc. Nat. Acad. Sci. 108, 467–472 (2011)
Imanuvilov O., Uhlmann G., Yamamoto M.: Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets. Inverse Probl. 27, 085007 (2011)
Kang H., Uhlmann G.: Inverse problem for the Pauli Hamiltonian in two dimensions. J. Fourier Anal. Appl. 10(2), 201–215 (2004)
Kobayashi, S.: Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan, 15. Kanô Memorial Lectures, 5. Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo (1987)
Lassas M., Uhlmann G.: On determining a Riemannian manifold from the Dirichlet-to-Neumann map. Ann. Sci. École Norm. Sup. (4) 34(5), 771–787 (2001)
Li X.: Inverse scattering problem for the Schroedinger operator with external Yang–Mills potentials in two dimensions at fixed energy. Comm. Partial Differ. Equ. 30(4–6), 451–482 (2005)
Ma, X., Marinescu, G.: Holomorphic Morse inequalities and Bergman kernels. Progress in Mathematics, 254. Birkhäuser Verlag, Basel (2007)
McDuff, D., Salamon, D.: J-holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications, 52. American Mathematical Society, Providence, RI (2004)
Nachman A.: Global uniqueness for a two dimensional inverse boundary value problem. Ann. Math 143, 71–96 (1996)
Newlander A., Nirenberg L.: Complex coordinates in almost complex manifolds. Ann. Math. 65, 391–404 (1957)
Novikov R.G., Santacesaria M.: Global uniqueness and reconstruction for the multi-channel Gel’fand-Calderón inverse problem in two dimensions. Bull. Sci. Math. 135(5), 421–434 (2011)
Schrader R., Taylor M.: Small h asymptotics for quantum partition functions associated to particles in external Yang–Mills potentials. Comm. Math. Phys. 92, 555–594 (1984)
Sylvester J.: An anisotropic inverse boundary value problem. Commun. Pure Appl. Math. 43, 201–232 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jan Derezinski.
P. A. was supported by a postdoctoral fellowship of the Foundation Sciences Mathématiques de Paris.
C.G. is supported by ANR Grants No. ANR-09-JCJC-0099-01 and ANR-10-BLAN 0105.
L.T. was supported by NSF Grant No. DMS-080750 G.U. was supported by NSF, a Chancellor Professorship at UC Berkeley and a Senior Clay Award.
We thank the anonymous referee for the careful reading of the article’s manuscript.
Rights and permissions
About this article
Cite this article
Albin, P., Guillarmou, C., Tzou, L. et al. Inverse Boundary Problems for Systems in Two Dimensions. Ann. Henri Poincaré 14, 1551–1571 (2013). https://doi.org/10.1007/s00023-012-0229-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-012-0229-1