×

Global uniqueness for a two-dimensional semilinear elliptic inverse problem. (English) Zbl 0849.35148

Authors’ abstract: For a general class of nonlinear Schrödinger equation \(- \Delta u+ a(x, u)= 0\) in a bounded planar domain \(\Omega\) we show that the function \(a(x, u)\) can be recovered from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary \(\partial \Omega\).

MSC:

35R30 Inverse problems for PDEs
35J10 Schrödinger operator, Schrödinger equation
35J25 Boundary value problems for second-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209 – 273. · Zbl 0459.60069 · doi:10.1002/cpa.3160350206
[2] Yu. M. Berezanskii, Trudy Moskov. Mat. Obshch. 7 (1958), 3-62.
[3] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224. · Zbl 0361.35003
[4] Victor Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations 92 (1991), no. 2, 305 – 316. · Zbl 0728.35141 · doi:10.1016/0022-0396(91)90051-A
[5] V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal. 124 (1993), no. 1, 1 – 12. · Zbl 0804.35150 · doi:10.1007/BF00392201
[6] Victor Isakov and Ziqi Sun, The inverse scattering at fixed energies in two dimensions, Indiana Univ. Math. J. 44 (1995), no. 3, 883 – 896. · Zbl 0855.35128 · doi:10.1512/iumj.1995.44.2013
[7] Victor Isakov and John Sylvester, Global uniqueness for a semilinear elliptic inverse problem, Comm. Pure Appl. Math. 47 (1994), no. 10, 1403 – 1410. · Zbl 0817.35126 · doi:10.1002/cpa.3160471005
[8] Adrian I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), no. 3, 531 – 576. · Zbl 0675.35084 · doi:10.2307/1971435
[9] Adrian I. Nachman, Inverse scattering at fixed energy, Mathematical physics, X (Leipzig, 1991) Springer, Berlin, 1992, pp. 434 – 441. · Zbl 0947.81562 · doi:10.1007/978-3-642-77303-7_48
[10] -, Global uniqueness for a two-dimensional inverse boundary value problem, Univ. of Rochester, Dept. of Math. Preprint Series 19, 1993;Ann. of Math. (1995) (to appear).
[11] R. G. Novikov, The inverse scattering problem on a fixed energy level for the two-dimensional Schrödinger operator, J. Funct. Anal. 103 (1992), no. 2, 409 – 463. · Zbl 0762.35077 · doi:10.1016/0022-1236(92)90127-5
[12] Zi Qi Sun, On an inverse boundary value problem in two dimensions, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1101 – 1113. · Zbl 0704.35132 · doi:10.1080/03605308908820646
[13] Ziqi Sun, On a quasilinear inverse boundary value problem, Math. Z. 221 (1996), no. 2, 293 – 305. · Zbl 0843.35137 · doi:10.1007/BF02622117
[14] Zi Qi Sun and Gunther Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J. 62 (1991), no. 1, 131 – 155. · Zbl 0728.35132 · doi:10.1215/S0012-7094-91-06206-X
[15] Zi Qi Sun and Gunther Uhlmann, Recovery of singularities for formally determined inverse problems, Comm. Math. Phys. 153 (1993), no. 3, 431 – 445. · Zbl 0795.35142
[16] John Sylvester and Gunther Uhlmann, A uniqueness theorem for an inverse boundary value problem in electrical prospection, Comm. Pure Appl. Math. 39 (1986), no. 1, 91 – 112. · Zbl 0611.35088 · doi:10.1002/cpa.3160390106
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.