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On a quasilinear inverse boundary value problem. (English) Zbl 0843.35137

We show that the Dirichlet to Neumann map associated to the quasilinear isotropic elliptic equation \(\nabla \cdot \gamma(x, u) \nabla u= 0\) determines uniquely the scalar coefficient \(\gamma(x, z)\), where \((x, z)\in \Omega\times \mathbb{R}\), \(\Omega\subset \mathbb{R}^n\) and \(n\geq 2\). This result generalizes a well-known global uniqueness theorem for an inverse boundary value problem for the linear isotropic elliptic equation \(\nabla\cdot \gamma(x) \nabla u= 0\) to quasilinear isotropic elliptic equations. We also consider the case of quasilinear anisotropic elliptic equations, where \(\gamma(x,z)\) is replaced by a positive definite matrix function \(A(x, z)\). We study an example in which we show that the Dirichlet to Neumann map determines the matrix coefficient \(A(x, z)\) modulo the group of diffeomorphisms which are the identity on the boundary of \(\Omega\).
Reviewer: Z.Sun (Wichita)

MSC:

35R30 Inverse problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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References:

[1] Calderon, A.: On an inverse boundary value problem, ”Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matematica, Rio de Janeiro, 1980”, 65–73
[2] Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Springer-Verlag, Berlin/New York, 1982 · Zbl 1042.35002
[3] Isakov, V.: On uniqueness in inverse problems for quasilinear parabolic equations, to appear in Arch. Rational Mech. Anal.
[4] Isakov, V., Sylvester, J.: Global uniqueness for a semilinear elliptic inverse problem, to appear in Comm. Pure Appl. Math. · Zbl 0817.35126
[5] Kohn, R., Vogelius, M.: Identification of an unknown conductivity by means of measurements at the boundary, in Inverse Problems. D.W. McLaughlin, ed., SIAM-AMS Proc. No.14, 113–123 (1984) · Zbl 0573.35084
[6] Lee, J., Uhlmann, G.: Determining anisotropic real-analytic conductivities by boundary measurements. Comm. Pure Appl. Math.42, 1097–1112 (1989) · Zbl 0702.35036 · doi:10.1002/cpa.3160420804
[7] Nachman, A.: Global uniqueness for a two-dimensional inverse boundary value problem. (preprint) · Zbl 0857.35135
[8] Sylvester, J.: An anisotropic inverse boundary value problem. Comm. Pure Appl. Math. Vol.XLIII, 201–232 (1990) · Zbl 0709.35102 · doi:10.1002/cpa.3160430203
[9] Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math.125, 153–169 (1987) · Zbl 0625.35078 · doi:10.2307/1971291
[10] Sylvester, J., Uhlmann, G.: Remarks on an inverse boundary value problem, in Pseudo-Differential Operators. Oberwolfach 1986, edited by H. O. Cordes, B. Gramsch and H. Widom, Lecture notes in Math.1256, 430–441
[11] Sylvester, J., Uhlmann, G.: The Dirichlet to Neumann map and applications, in ”Inverse Problems in Partial Differential Equations”, SIAM Proc. Series, Philadelphia 1990 · Zbl 0713.35100
[12] Sylvester, J., Uhlmann, G.: Inverse problems in anisotropic media. Inverse Scattering and Applications. Contemp. Math.122, AMS, Providence, RI 105–117 (1991) · Zbl 0748.35057
[13] Sun, Z.: The inverse conductivity problem in two dimensions. J. Diff. Eq.87, 227–255 (1990) · Zbl 0716.35080 · doi:10.1016/0022-0396(90)90002-7
[14] Sun, Z., Uhlmann, G.: Generic uniqueness for an inverse boundary value problem. Duke Math. J.62(1), 131–155 (1991) · Zbl 0728.35132 · doi:10.1215/S0012-7094-91-06206-X
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