Inverse problems for semilinear elliptic PDE with measurements at a single point

Salo, Mikko; Tzou, Leo

doi:10.1090/proc/16255

Inverse problems for semilinear elliptic PDE with measurements at a single point
HTML articles powered by AMS MathViewer

by Mikko Salo and Leo Tzou;

Proc. Amer. Math. Soc. 151 (2023), 2023-2030

DOI: https://doi.org/10.1090/proc/16255

Published electronically: February 10, 2023

HTML | PDF | Request permission

Abstract:

We consider the inverse problem of determining a potential in a semilinear elliptic equation from the knowledge of the Dirichlet-to-Neumann map. For bounded Euclidean domains we prove that the potential is uniquely determined by the Dirichlet-to-Neumann map measured at a single boundary point, or integrated against a fixed measure. This result is valid even when the Dirichlet data is only given on a small subset of the boundary. We also give related uniqueness results on Riemannian manifolds.

References

Yernat M. Assylbekov and Ting Zhou, Direct and inverse problems for the nonlinear time-harmonic Maxwell equations in Kerr-type media, J. Spectr. Theory 11 (2021), no. 1, 1–38. MR 4233204, DOI 10.4171/jst/334
Amin Boumenir and Vu Kim Tuan, Reconstructing the wave speed and the source, Math. Methods Appl. Sci. 44 (2021), no. 18, 14470–14480. MR 4342848, DOI 10.1002/mma.7713
Cătălin I. Cârstea and Ali Feizmohammadi, An inverse boundary value problem for certain anisotropic quasilinear elliptic equations, J. Differential Equations 284 (2021), 318–349. MR 4227095, DOI 10.1016/j.jde.2021.02.044
Cătălin I. Cârstea, Ali Feizmohammadi, Yavar Kian, Katya Krupchyk, and Gunther Uhlmann, The Calderón inverse problem for isotropic quasilinear conductivities, Adv. Math. 391 (2021), Paper No. 107956, 31. MR 4300916, DOI 10.1016/j.aim.2021.107956
Cătălin I. Cârstea, Tuhin Ghosh, and Gunther Uhlmann, An inverse problem for the porous medium equation with partial data and a possibly singular absorption term, arXiv:2108.12970 (2021).
Cătălin I. Cârstea, Gen Nakamura, and Manmohan Vashisth, Reconstruction for the coefficients of a quasilinear elliptic partial differential equation, Appl. Math. Lett. 98 (2019), 121–127. MR 3964222, DOI 10.1016/j.aml.2019.06.009
David Dos Santos Ferreira, Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, On the linearized local Calderón problem, Math. Res. Lett. 16 (2009), no. 6, 955–970. MR 2576684, DOI 10.4310/MRL.2009.v16.n6.a4
Ali Feizmohammadi and Lauri Oksanen, An inverse problem for a semi-linear elliptic equation in Riemannian geometries, J. Differential Equations 269 (2020), no. 6, 4683–4719. MR 4104456, DOI 10.1016/j.jde.2020.03.037
Colin Guillarmou, Mikko Salo, and Leo Tzou, The linearized Calderón problem on complex manifolds, Acta Math. Sin. (Engl. Ser.) 35 (2019), no. 6, 1043–1056. MR 3952702, DOI 10.1007/s10114-019-8129-7
Fengbo Hang, Xiaodong Wang, and Xiaodong Yan, An integral equation in conformal geometry, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 1, 1–21. MR 2483810, DOI 10.1016/j.anihpc.2007.03.006
Victor Isakov and Adrian I. Nachman, Global uniqueness for a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3375–3390. MR 1311909, DOI 10.1090/S0002-9947-1995-1311909-1
Victor Isakov and John Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math. 47 (1994), no. 10, 1403–1410. MR 1295934, DOI 10.1002/cpa.3160471005
V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal. 124 (1993), no. 1, 1–12. MR 1233645, DOI 10.1007/BF00392201
Yavar Kian, Katya Krupchyk, and Gunther Uhlmann, Partial data inverse problems for quasilinear conductivity equations, Math. Ann. (2022), https://doi.org/10.1007/s00208-022-02367-y.
Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann, Inverse problems for Lorentzian manifolds and non-linear hyperbolic equations, Invent. Math. 212 (2018), no. 3, 781–857. MR 3802298, DOI 10.1007/s00222-017-0780-y
Kyeonbae Kang and Gen Nakamura, Identification of nonlinearity in a conductivity equation via the Dirichlet-to-Neumann map, Inverse Problems 18 (2002), no. 4, 1079–1088. MR 1929283, DOI 10.1088/0266-5611/18/4/309
Steven G. Krantz, Calculation and estimation of the Poisson kernel, J. Math. Anal. Appl. 302 (2005), no. 1, 143–148. MR 2107352, DOI 10.1016/j.jmaa.2004.08.010
Katya Krupchyk and Gunther Uhlmann, Partial data inverse problems for semilinear elliptic equations with gradient nonlinearities, Math. Res. Lett. 27 (2020), no. 6, 1801–1824. MR 4216606, DOI 10.4310/MRL.2020.v27.n6.a10
Katya Krupchyk and Gunther Uhlmann, A remark on partial data inverse problems for semilinear elliptic equations, Proc. Amer. Math. Soc. 148 (2020), no. 2, 681–685. MR 4052205, DOI 10.1090/proc/14844
Katya Krupchyk and Gunther Uhlmann, Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds, Anal. PDE (2022), arXiv:2009.05089.
Nicolas Lerner, Carleman inequalities, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 353, Springer, Cham, 2019. An introduction and more. MR 3932103, DOI 10.1007/978-3-030-15993-1
Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin, and Mikko Salo, Inverse problems for elliptic equations with power type nonlinearities, J. Math. Pures Appl. (9) 145 (2021), 44–82 (English, with English and French summaries). MR 4188325, DOI 10.1016/j.matpur.2020.11.006
Matti Lassas, Tony Liimatainen, Yi-Hsuan Lin, and Mikko Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam. 37 (2021), no. 4, 1553–1580. MR 4269409, DOI 10.4171/rmi/1242
Matti Lassas, Tony Liimatainen, Leyter Potenciano-Machado, and Teemu Tyni, Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation, J. Differential Equations 337 (2022), 395–435. MR 4473034, DOI 10.1016/j.jde.2022.08.010
Tony Liimatainen, Yi-Hsuan Lin, Mikko Salo, and Teemu Tyni, Inverse problems for elliptic equations with fractional power type nonlinearities, J. Differential Equations 306 (2022), 189–219. MR 4332042, DOI 10.1016/j.jde.2021.10.015
Ru-Yu Lai and Ting Zhou, Partial data inverse problems for nonlinear magnetic Schrödinger equations, arXiv:2007.02475 (2020).
Ziqi Sun and Gunther Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math. 119 (1997), no. 4, 771–797. MR 1465069, DOI 10.1353/ajm.1997.0027
Ziqi Sun, On a quasilinear inverse boundary value problem, Math. Z. 221 (1996), no. 2, 293–305. MR 1376299, DOI 10.1007/BF02622117
Leo Tzou, Determining Riemannian manifolds from nonlinear wave observations at a single point, arXiv:2102.01841 (2021).