The Dirichlet to Neumann operator for elliptic complexes

Tarkhanov, N.

doi:10.1090/S0002-9947-2011-05460-7

The Dirichlet to Neumann operator for elliptic complexes
HTML articles powered by AMS MathViewer

by N. Tarkhanov

Trans. Amer. Math. Soc. 363 (2011), 6421-6437

DOI: https://doi.org/10.1090/S0002-9947-2011-05460-7

Published electronically: July 22, 2011

PDF | Request permission

Abstract:

We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary.

References

M. I. Belishev, The Calderon problem for two-dimensional manifolds by the BC-method, SIAM J. Math. Anal. 35 (2003), no. 1, 172–182. MR 2001471, DOI 10.1137/S0036141002413919
M. I. Belishev, Some remarks on the impedance tomography problem for 3d-manifolds, Cubo 7 (2005), no. 1, 43–55 (English, with English and Spanish summaries). MR 2140807
Mikhail Belishev and Vladimir Sharafutdinov, Dirichlet to Neumann operator on differential forms, Bull. Sci. Math. 132 (2008), no. 2, 128–145. MR 2387822, DOI 10.1016/j.bulsci.2006.11.003
Mourad Bellassoued and Mourad Choulli, Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map, J. Funct. Anal. 258 (2010), no. 1, 161–195. MR 2557958, DOI 10.1016/j.jfa.2009.06.010
Nuutti Hyvönen, Comparison of idealized and electrode Dirichlet-to-Neumann maps in electric impedance tomography with an application to boundary determination of conductivity, Inverse Problems 25 (2009), no. 8, 085008, 18. MR 2529198, DOI 10.1088/0266-5611/25/8/085008
Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567–591. MR 2299741, DOI 10.4007/annals.2007.165.567
J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492. MR 181815, DOI 10.1002/cpa.3160180305
Matti Lassas and Gunther Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 771–787 (English, with English and French summaries). MR 1862026, DOI 10.1016/S0012-9593(01)01076-X
Matti Lassas, Michael Taylor, and Gunther Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), no. 2, 207–221. MR 2014876, DOI 10.4310/CAG.2003.v11.n2.a2
John M. Lee and Gunther Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), no. 8, 1097–1112. MR 1029119, DOI 10.1002/cpa.3160420804
Charles B. Morrey Jr., A variational method in the theory of harmonic integrals. II, Amer. J. Math. 78 (1956), 137–170. MR 87765, DOI 10.2307/2372488
Adrian I. Nachman, Lassi Päivärinta, and Ari Teirilä, On imaging obstacles inside inhomogeneous media, J. Funct. Anal. 252 (2007), no. 2, 490–516. MR 2360925, DOI 10.1016/j.jfa.2007.06.020
Adrian Nachman, Alexandru Tamasan, and Alexandre Timonov, Recovering the conductivity from a single measurement of interior data, Inverse Problems 25 (2009), no. 3, 035014, 16. MR 2480184, DOI 10.1088/0266-5611/25/3/035014
Yakov Roitberg, Elliptic boundary value problems in the spaces of distributions, Mathematics and its Applications, vol. 384, Kluwer Academic Publishers Group, Dordrecht, 1996. Translated from the Russian by Peter Malyshev and Dmitry Malyshev. MR 1423135, DOI 10.1007/978-94-011-5410-9
Martin Schechter, Negative norms and boundary problems, Ann. of Math. (2) 72 (1960), 581–593. MR 125333, DOI 10.2307/1970230
Martin Schechter, Principles of functional analysis, 2nd ed., Graduate Studies in Mathematics, vol. 36, American Mathematical Society, Providence, RI, 2002. MR 1861991, DOI 10.1090/gsm/036
B.-W. Schulze, A. Shlapunov, and N. Tarkhanov, Green integrals on manifolds with cracks, Ann. Global Anal. Geom. 24 (2003), no. 2, 131–160. MR 1990112, DOI 10.1023/A:1024458312097
Nikolai N. Tarkhanov, Complexes of differential operators, Mathematics and its Applications, vol. 340, Kluwer Academic Publishers Group, Dordrecht, 1995. Translated from the 1990 Russian original by P. M. Gauthier and revised by the author. MR 1368856, DOI 10.1007/978-94-011-0327-5
Michael E. Taylor, Partial differential equations. II, Applied Mathematical Sciences, vol. 116, Springer-Verlag, New York, 1996. Qualitative studies of linear equations. MR 1395149, DOI 10.1007/978-1-4757-4187-2