Global uniqueness for a two-dimensional semilinear elliptic inverse problem

Authors:
Victor Isakov and Adrian I. Nachman

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

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Abstract | References | Similar Articles | Additional Information

Abstract: For a general class of nonlinear Schrödinger equations in a bounded planar domain we show that the function can be recovered from knowledge of the corresponding Dirichlet-to-Neumann map on the boundary .

**[A-S]**M. Aizenman and B. Simon,*Brownian motion and Harnack's inequality for Schrödinger operators*, Comm. Pure Appl. Math.**35**(1982), 209-273. MR**644024 (84a:35062)****[B]**Yu. M. Berezanskii, Trudy Moskov. Mat. Obshch.**7**(1958), 3-62.**[G-T]**D. Gilbarg and N. Trudinger,*Elliptic partial differential equations of second order*, Springer-Verlag, 1977. MR**0473443 (57:13109)****[Is 1]**V. Isakov,*Completeness of products of solutions and some inverse problems for PDE*, J. Differential Equations**92**(1991), 305-317. MR**1120907 (92g:35044)****[Is 2]**-,*On uniqueness in inverse problems for semilinear parabolic equations*, Arch. Rational Mech. Anal.**124**(1993), 1-13. MR**1233645 (94h:35263)****[Is-Su]**V. Isakov and Z. Sun,*The inverse scattering at fixed energies in two dimensions*, Indiana Univ. Math. J. (1995) (to appear). MR**1375354 (96m:35334)****[Is-Sy]**V. Isakov and J. Sylvester,*Global uniqueness for a semilinear elliptic inverse problem*, Comm. Pure Appl. Math**47**(1994), 1403-1410. MR**1295934 (95h:35243)****[N 1]**A.I. Nachman,*Reconstructions from boundary measurements*, Ann. of Math.**128**(1988), 531-577. MR**970610 (90i:35283)****[N 2]**-,*Inverse scattering at fixed energy*, Proc. 10th International Congress on Math. Phys., Leipzig 1991, edited by K. Schmüdgen, Springer-Verlag, 1992, pp. 434-441. MR**1386440****[N 3]**-,*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).**[No]**R. G. Novikov,*The inverse scattering problem on a fixed energy level for the two dimensional Schrödinger operator*, J. Funct. Anal**103**(1992), 409-463. MR**1151554 (93e:35080)****[S 1]**Z. Sun,*On an inverse boundary value problem in two dimensions*, Comm. Partial Differential Equations**14**(1989), 1101-1113. MR**1017066 (91i:35210)****[S 2]**-,*On a quasilinear inverse boundary value problem*, Math. Z. (1995) (to appear). MR**1376299 (96m:35109)****[Su-U1]**Z. Sun and G. Uhlmann,*Generic uniqueness for an inverse boundary value problem*, Duke Math. J.**62**(1991), 131-155. MR**1104326 (92b:35172)****[Su-U2]**-,*Recovery of singularities for formally determined inverse problems*, Comm. Math. Phys.**153**(1993), 431-445. MR**1218927 (94f:35146)****[Sy-U]**J. Sylvester and G. Uhlmann,*A uniqueness theorem for an inverse boundary value problem in electrical prospection*, Comm. Pure Appl. Math**39**(1986), 92-112. MR**820341 (87j:35377)**

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1311909-1

Article copyright:
© Copyright 1995
American Mathematical Society