Global uniqueness for a two-dimensional semilinear elliptic inverse problem
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- by Victor Isakov and Adrian I. Nachman PDF
- Trans. Amer. Math. Soc. 347 (1995), 3375-3390 Request permission
Abstract:
For a general class of nonlinear Schrödinger equations $- \Delta u + a(x,u) = 0$ 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 $\partial \Omega$.References
- M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209–273. MR 644024, DOI 10.1002/cpa.3160350206 Yu. M. Berezanskii, Trudy Moskov. Mat. Obshch. 7 (1958), 3-62.
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
- Victor Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations 92 (1991), no. 2, 305–316. MR 1120907, DOI 10.1016/0022-0396(91)90051-A
- 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
- Victor Isakov and Ziqi Sun, The inverse scattering at fixed energies in two dimensions, Indiana Univ. Math. J. 44 (1995), no. 3, 883–896. MR 1375354, DOI 10.1512/iumj.1995.44.2013
- 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
- Adrian I. Nachman, Reconstructions from boundary measurements, Ann. of Math. (2) 128 (1988), no. 3, 531–576. MR 970610, DOI 10.2307/1971435
- Adrian I. Nachman, Inverse scattering at fixed energy, Mathematical physics, X (Leipzig, 1991) Springer, Berlin, 1992, pp. 434–441. MR 1386440, DOI 10.1007/978-3-642-77303-7_{4}8 —, 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).
- 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. MR 1151554, DOI 10.1016/0022-1236(92)90127-5
- Zi Qi Sun, On an inverse boundary value problem in two dimensions, Comm. Partial Differential Equations 14 (1989), no. 8-9, 1101–1113. MR 1017066, DOI 10.1080/03605308908820646
- Ziqi Sun, On a quasilinear inverse boundary value problem, Math. Z. 221 (1996), no. 2, 293–305. MR 1376299, DOI 10.1007/BF02622117
- Zi Qi Sun and Gunther Uhlmann, Generic uniqueness for an inverse boundary value problem, Duke Math. J. 62 (1991), no. 1, 131–155. MR 1104326, DOI 10.1215/S0012-7094-91-06206-X
- Zi Qi Sun and Gunther Uhlmann, Recovery of singularities for formally determined inverse problems, Comm. Math. Phys. 153 (1993), no. 3, 431–445. MR 1218927
- 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. MR 820341, DOI 10.1002/cpa.3160390106
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 3375-3390
- MSC: Primary 35R30; Secondary 35J60
- DOI: https://doi.org/10.1090/S0002-9947-1995-1311909-1
- MathSciNet review: 1311909