Local uniqueness for an inverse boundary value problem with partial data

Harrach, Bastian; Ullrich, Marcel

doi:10.1090/proc/12991

Local uniqueness for an inverse boundary value problem with partial data
HTML articles powered by AMS MathViewer

by Bastian Harrach and Marcel Ullrich PDF

Proc. Amer. Math. Soc. 145 (2017), 1087-1095 Request permission

Abstract:

In dimension $n\geq 3$, we prove a local uniqueness result for the potentials $q$ of the Schrödinger equation $-\Delta u+qu=0$ from partial boundary data. More precisely, we show that potentials $q_1,q_2\in L^\infty$ with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where $q_1\geq q_2$ and $q_1\not \equiv q_2$.

References

Kari Astala and Lassi Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), no. 1, 265–299. MR 2195135, DOI 10.4007/annals.2006.163.265
Nicolas Bourbaki, Elements of mathematics: Topological vector spaces, chapters 1-5, Springer-Verlag, 2003.
A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), no. 1, 19–33. MR 2387648, DOI 10.1515/jiip.2008.002
Alberto-P. Calderón, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980) Soc. Brasil. Mat., Rio de Janeiro, 1980, pp. 65–73. MR 590275
Alberto P. Calderón, On an inverse boundary value problem, Comput. Appl. Math. 25 (2006), no. 2-3, 133–138. MR 2321646, DOI 10.1590/S0101-82052006000200002
Djairo G. de Figueiredo and Jean-Pierre Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations 17 (1992), no. 1-2, 339–346. MR 1151266, DOI 10.1080/03605309208820844
Florian Frühauf, Bastian Gebauer, and Otmar Scherzer, Detecting interfaces in a parabolic-elliptic problem from surface measurements, SIAM J. Numer. Anal. 45 (2007), no. 2, 810–836. MR 2300298, DOI 10.1137/050641545
Bastian Gebauer, Localized potentials in electrical impedance tomography, Inverse Probl. Imaging 2 (2008), no. 2, 251–269. MR 2395143, DOI 10.3934/ipi.2008.2.251
Boaz Haberman and Daniel Tataru, Uniqueness in Calderón’s problem with Lipschitz conductivities, Duke Math. J. 162 (2013), no. 3, 496–516. MR 3024091, DOI 10.1215/00127094-2019591
Islam Eddine Hadi and N. Tsouli, Strong unique continuation of eigenfunctions for $p$-Laplacian operator, Int. J. Math. Math. Sci. 25 (2001), no. 3, 213–216. MR 1812385, DOI 10.1155/S0161171201004744
Bastian Harrach, On uniqueness in diffuse optical tomography, Inverse Problems 25 (2009), no. 5, 055010, 14. MR 2501028, DOI 10.1088/0266-5611/25/5/055010
Bastian Harrach, Simultaneous determination of the diffusion and absorption coefficient from boundary data, Inverse Probl. Imaging 6 (2012), no. 4, 663–679. MR 2998710, DOI 10.3934/ipi.2012.6.663
Bastian Harrach and Marcel Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography, SIAM J. Math. Anal. 45 (2013), no. 6, 3382–3403. MR 3126995, DOI 10.1137/120886984
Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1994. Pseudo-differential operators; Corrected reprint of the 1985 original. MR 1313500
M. Ikehata, Size estimation of inclusion, J. Inverse Ill-Posed Probl. 6 (1998), no. 2, 127–140. MR 1637360, DOI 10.1515/jiip.1998.6.2.127
Oleg Yu. Imanuvilov, Gunther Uhlmann, and Masahiro Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc. 23 (2010), no. 3, 655–691. MR 2629983, DOI 10.1090/S0894-0347-10-00656-9
Victor Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging 1 (2007), no. 1, 95–105. MR 2262748, DOI 10.3934/ipi.2007.1.95
Hyeonbae Kang, Jin Keun Seo, and Dongwoo Sheen, The inverse conductivity problem with one measurement: stability and estimation of size, SIAM J. Math. Anal. 28 (1997), no. 6, 1389–1405. MR 1474220, DOI 10.1137/S0036141096299375
Carlos Kenig and Mikko Salo, The Calderón problem with partial data on manifolds and applications, Anal. PDE 6 (2013), no. 8, 2003–2048. MR 3198591, DOI 10.2140/apde.2013.6.2003
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
C. E. Kenig and Mikko Salo, Recent progress in the calderón problem with partial data, Contemp. Math 615 (2014), 193–222.
Robert Kohn and Michael Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math. 37 (1984), no. 3, 289–298. MR 739921, DOI 10.1002/cpa.3160370302
R. V. Kohn and M. Vogelius, Determining conductivity by boundary measurements. II. Interior results, Comm. Pure Appl. Math. 38 (1985), no. 5, 643–667. MR 803253, DOI 10.1002/cpa.3160380513
Adrian Nachman and Brian Street, Reconstruction in the Calderón problem with partial data, Comm. Partial Differential Equations 35 (2010), no. 2, 375–390. MR 2748629, DOI 10.1080/03605300903296322
Adrian I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71–96. MR 1370758, DOI 10.2307/2118653
Rachid Regbaoui, Unique continuation from sets of positive measure, Carleman estimates and applications to uniqueness and control theory (Cortona, 1999) Progr. Nonlinear Differential Equations Appl., vol. 46, Birkhäuser Boston, Boston, MA, 2001, pp. 179–190. MR 1839175
John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380, DOI 10.2307/1971291