doi: 10.3934/ipi.2022008
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A uniqueness theorem for inverse problems in quasilinear anisotropic media

Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260, USA

Received  December 2021 Revised  January 2022 Early access March 2022

We study the question of whether one can uniquely determine a scalar quasilinear conductivity in an anisotropic medium by making voltage and current measurements at the boundary. This paper is dedicated to the memory of Professor Victor Isakov, who has made enormous contribution to the theory of inverse problem.

Citation: Md. Ibrahim Kholil, Ziqi Sun. A uniqueness theorem for inverse problems in quasilinear anisotropic media. Inverse Problems and Imaging, doi: 10.3934/ipi.2022008
References:
[1]

L. Ahlofors, Quasiconformal Mappings, Van Nostrand Co., Inc., Toronto, Ont.-New York-London 1966.

[2]

J. Ferrand, Geometrical interpretation of scalar curvature and regularity of conformal home-omorphisms, Differential Geometry and Relativity, 3 (1976), 91-105.  doi: 10.1007/978-94-010-1508-0_11.

[3]

J. Ferrand, The action of conformal transformations on a riemannian manifold, Math. Ann., 304 (1996), 277-291.  doi: 10.1007/BF01446294.

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977. doi: https://doi.org/10.1007/978-3-642-96379-7.

[5]

V. Isakov, On uniqueness in inverse problems for quasilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12.  doi: 10.1007/BF00392201.

[6]

V. Isakov and A. Nachman, Global uniqueness in a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc., 347 (1995), 3375-3390.  doi: 10.1090/S0002-9947-1995-1311909-1.

[7]

V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math., 47 (1994), 1403-1410.  doi: 10.1002/cpa.3160471005.

[8]

Y. Kian, K. Krupchyk and G. Uhlmann, Partial data inverse problems for quasilinear conductivity equations, preprint, arXiv: 2010.11409.

[9]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.  doi: 10.1002/cpa.3160420804.

[10]

C. Munoz and G. Uhlmann, The Calderón problem for quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1143-1166.  doi: 10.1016/j.anihpc.2020.03.004.

[11]

S. Myers and N. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math., 40 (1939), 400-416.  doi: 10.2307/1968928.

[12]

Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.  doi: 10.1007/BF02622117.

[13]

Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797.  doi: 10.1353/ajm.1997.0027.

[14]

J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201-232.  doi: 10.1002/cpa.3160430203.

show all references

References:
[1]

L. Ahlofors, Quasiconformal Mappings, Van Nostrand Co., Inc., Toronto, Ont.-New York-London 1966.

[2]

J. Ferrand, Geometrical interpretation of scalar curvature and regularity of conformal home-omorphisms, Differential Geometry and Relativity, 3 (1976), 91-105.  doi: 10.1007/978-94-010-1508-0_11.

[3]

J. Ferrand, The action of conformal transformations on a riemannian manifold, Math. Ann., 304 (1996), 277-291.  doi: 10.1007/BF01446294.

[4]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1977. doi: https://doi.org/10.1007/978-3-642-96379-7.

[5]

V. Isakov, On uniqueness in inverse problems for quasilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12.  doi: 10.1007/BF00392201.

[6]

V. Isakov and A. Nachman, Global uniqueness in a two-dimensional semilinear elliptic inverse problem, Trans. Amer. Math. Soc., 347 (1995), 3375-3390.  doi: 10.1090/S0002-9947-1995-1311909-1.

[7]

V. Isakov and J. Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math., 47 (1994), 1403-1410.  doi: 10.1002/cpa.3160471005.

[8]

Y. Kian, K. Krupchyk and G. Uhlmann, Partial data inverse problems for quasilinear conductivity equations, preprint, arXiv: 2010.11409.

[9]

J. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112.  doi: 10.1002/cpa.3160420804.

[10]

C. Munoz and G. Uhlmann, The Calderón problem for quasilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), 1143-1166.  doi: 10.1016/j.anihpc.2020.03.004.

[11]

S. Myers and N. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math., 40 (1939), 400-416.  doi: 10.2307/1968928.

[12]

Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305.  doi: 10.1007/BF02622117.

[13]

Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797.  doi: 10.1353/ajm.1997.0027.

[14]

J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math., 43 (1990), 201-232.  doi: 10.1002/cpa.3160430203.

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