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SEMILINEAR CALDERÓN PROBLEM ON STEIN MANIFOLDS WITH KÄHLER METRIC

Part of: Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  20 May 2020

YILIN MA Open the ORCID record for YILIN MA [Opens in a new window]  and
LEO TZOU Open the ORCID record for LEO TZOU [Opens in a new window]
YILIN MA*
Affiliation:
School of Mathematics and Statistics,The University of Sydney, Camperdown, New South Wales 2006, Australia email K.Ma@maths.usyd.edu.au
LEO TZOU
Affiliation:
School of Mathematics and Statistics,The University of Sydney, Camperdown, New South Wales 2006, Australia email leo.tzou@sydney.edu.au
*
✉K.Ma@maths.usyd.edu.au
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Abstract

We extend existing methods which treat the semilinear Calderón problem on a bounded domain to a class of complex manifolds with Kähler metric. Given two semilinear Schrödinger operators with the same Dirchlet-to-Neumann data, we show that the integral identities that appear naturally in the determination of the analytic potentials are enough to deduce uniqueness on the boundary up to infinite order. By exploiting the assumed complex structure, this information allows us to apply the method of stationary phase and recover the potentials in the interior as well.

Keywords

nonlinear inverse problemspartial dataCalderón problem

MSC classification

Primary: 35R30: Inverse problems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 1 , February 2021 , pp. 132 - 144
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

Angulo-Ardoy, P., ‘On the set of metrics without local limiting Carleman weights’, Inverse Probl. Imaging 11(1) (2017), 47–64.10.3934/ipi.2017003CrossRefGoogle Scholar
Angulo-Ardoy, P., Faraco, D. and Guijarro, L., ‘Sufficient conditions for the existence of limiting Carleman weights’, Forum Math. Sigma 5 (2017), Article E7.10.1017/fms.2017.4CrossRefGoogle Scholar
Brown, R. M., ‘Recovering the conductivity at the boundary from the Dirichlet to Neumann map: a pointwise result’, J. Inverse Ill-posed Probl. 9(6) (2001), 567–574.10.1515/jiip.2001.9.6.567CrossRefGoogle Scholar
Bukhgeim, A. L., ‘Recovering a potential from Cauchy data in the two-dimensional case’, J. Inverse Ill-posed Probl. 16(1) (2008), 19–35.10.1515/jiip.2008.002CrossRefGoogle Scholar
Feizmohammadi, A. and Oksanen, L., ‘An inverse problem for a semi-linear elliptic equation in Riemannian geometries’, Preprint, 2019, arXiv:1904.00608.10.1016/j.jde.2020.03.037CrossRefGoogle Scholar
Ferreira, D. D. S., Kenig, C. E., Salo, M. and Uhlmann, G., ‘Limiting Carleman weights and anisotropic inverse problems’, Invent. Math. 178(1) (2009), 119–171.10.1007/s00222-009-0196-4CrossRefGoogle Scholar
Ferreira, D. D. S., Kenig, C. E., Sjöstrand, J. and Uhlman, G., ‘On the linearized local Calderón problem’, Math. Res. Lett. 16(6) (2009), 966–970.10.4310/MRL.2009.v16.n6.a4CrossRefGoogle Scholar
Ferreira, D. D. S., Kurylev, Y., Lassas, M., Liimatainen, T. and Salo, M., ‘The linearized Calderón problem in transversally anisotropic geometries’, Int. Math. Res. Not. (to appear). Published online (23 October 2018).Google Scholar
Ferreira, D. D. S., Kurylev, Y., Lassas, M. and Salo, M., ‘The Calderón problem in transversally anisotropic geometries’, J. Eur. Math. Soc. 18(11) (2016), 2579–2626.10.4171/JEMS/649CrossRefGoogle Scholar
Guillarmou, C. and Tzou, L., ‘Calderón inverse problem with partial data on Riemann surfaces’, Duke Math. J. 158(1) (2011), 83–120.10.1215/00127094-1276310CrossRefGoogle Scholar
Guillarmou, C., Salo, M. and Tzou, L., ‘The linearized Calderón problem on complex manifolds’, Acta Math. Sin. (Engl. Ser.) 35(6) (2019), 1043–1056.10.1007/s10114-019-8129-7CrossRefGoogle Scholar
Imanuvilov, O. and Yamamoto, M., ‘Unique determination of potentials and semilinear terms of semilinear elliptic equations from partial Cauchy data’, J. Inverse Ill-Posed Probl. 21(1) (2013), 85–108.10.1515/jip-2012-0033CrossRefGoogle Scholar
Isakov, V. and Nachman, A. I., ‘Global uniqueness for a two-dimensional semilinear elliptic inverse problem’, Trans. Amer. Math. Soc. 347(9) (1995), 3375–3390.10.1090/S0002-9947-1995-1311909-1CrossRefGoogle Scholar
Isakov, V. and Sylvester, J., ‘Global uniqueness for a semilinear elliptic inverse problem’, Comm. Pure Appl. Math. 47(10) (1994), 1403–1410.10.1002/cpa.3160471005CrossRefGoogle Scholar
Kenig, C. E. and Salo, M., ‘Recent progress in the Calderón problem with partial data’, in: Inverse Problems and Applications Contemporary Mathematics, Contemporary Mathematics, Vol. 615 (American Mathematical Society, Providence, RI, 2014), 193–2-22.Google Scholar
Kenig, C. E., Salo, M. and Uhlmann, G., ‘Reconstructions from boundary measurements on admissible manifolds’, Inverse Probl. Imaging 5(4) (2011), 859–877.10.3934/ipi.2011.5.859CrossRefGoogle Scholar
Krupchyk, K. and Uhlmann, G., ‘Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities’, Preprint, 2019, arXiv:1909.08122.10.1090/proc/14844CrossRefGoogle Scholar
Krupchyk, K. and Uhlmann, G., ‘A remark on partial data inverse problems for semilinear elliptic equations’, Proc. Amer. Math. Soc. 148 (2020), 681–685.10.1090/proc/14844CrossRefGoogle Scholar
Lassas, M., Liimatainen, T., Lin, Y.-H and Salo, M., ‘Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations’, Preprint, 2019, arXiv:1905.02764.Google Scholar
Lassas, M., Liimatainen, T., Lin, Y.-H. and Salo, M., ‘Inverse problems for elliptic equations with power type nonlinearities’, Preprint, 2019, arXiv:1903.12562.Google Scholar
Nachman, A. I., ‘Global uniqueness for a two-dimensional inverse boundary value problem’, Ann. of Math. 143(1) (1996), 71–96.10.2307/2118653CrossRefGoogle Scholar
Nakamura, G. and Tanuma, K., ‘Local determination of conductivity at the boundary from the Dirichlet-to-Neumann map’, Inverse Problems 17(3) (2001), 405–419.10.1088/0266-5611/17/3/303CrossRefGoogle Scholar
Sjöstrand, J. and Uhlmann, G., ‘Local analytic regularity in the linearized Calderón problem’, Anal. PDE 9(3) (2016), 515–544.10.2140/apde.2016.9.515CrossRefGoogle Scholar