We show that the knowledge of Dirichlet to Neumann map for rough $ A $ and $ q $ in $ (-\Delta)^m +A\cdot D +q $ for $ m \geq 2 $ for a bounded domain in $ \mathbb{R}^n $, $ n \geq 3 $ determines $ A $ and $ q $ uniquely. This unique identifiability is proved via construction of complex geometrical optics solutions with sufficient decay of remainder terms, by using property of products of functions in Sobolev spaces.
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[1] | R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] | Y. M. Assylbekov, Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order, Inverse Problems, 32 (2016), 105009, 22pp. doi: 10.1088/0266-5611/32/10/105009. |
[3] | Y. M. Assylbekov and Y. Yang, Determining the first order perturbation of a polyharmonic operator on admissible manifolds, Journal of Differential Equations, 262 (2017), 590-614. doi: 10.1016/j.jde.2016.09.039. |
[4] | A. Behzadan and N. Holst, Multiplication in Sobolev Spaces, Revisited, arXiv: 1512.07379. |
[5] | R. Brown, Global uniqueness in the impedance-imaging problem for less regular conductivities, SIAM J. Math. Anal., 27 (1996), 1049-1056. doi: 10.1137/S0036141094271132. |
[6] | A. P. Calderon, On an inverse boundary value problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Rio de Janeiro, (1980), 65–73. |
[7] | P. Caro and K. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum of Mathematics, Pi, 4 (2016), e2, 28 pp. doi: 10.1017/fmp.2015.9. |
[8] | S. Chanillo, A problem in electrical prospection and an $n$-dimensional Borg–Levinson theorem, Proc. Amer. Math. Soc., 108 (1990), 761-767. doi: 10.2307/2047798. |
[9] | L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035. |
[10] | G. Folland, Real Analysis, Modern Techniques and their Applications, John Wiley & Sons, New York, 1984. |
[11] | F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3. |
[12] | A. Greenleaf, M. Lassas and G. Uhlmann, The Calderón problem for conormal potentials. Ⅰ. Global uniqueness and reconstruction, Comm. Pure Appl. Math., 56 (2003), 328-352. doi: 10.1002/cpa.10061. |
[13] | G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, 252. Springer, New York, 2009. |
[14] | B. Haberman, Unique determination of a magnetic Schrdinger operator with unbounded magnetic potential from boundary data, arXiv: 1512.01580. |
[15] | B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516. doi: 10.1215/00127094-2019591. |
[16] | B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm, Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3. |
[17] | M. Ikehata, A special Green's function for the biharmonic operator and its application to an inverse boundary value problem, Comput. Math. Appl., 22 (1991), 53-66. doi: 10.1016/0898-1221(91)90131-M. |
[18] | V. Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316. doi: 10.1016/0022-0396(91)90051-A. |
[19] | C. Kenig, J. Sjöstrand and G. Uhlmann, Carleman estimates and inverse problems for Dirac operators, Ann. Math., 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567. |
[20] | K. Krupchyk, M. Lassas and G. Uhlmannf, Inverse Boundary value Problems for the Perturbed Polyharmonic Operator, Transactions AMS, 366 (2014), 95-112. doi: 10.1090/S0002-9947-2013-05713-3. |
[21] | K. Krupchyk and G. Uhlmann, Uniqueness in an inverse boundary problem for a magnetic Schrödinger operator with a bounded magnetic potential, Comm. Math. Phys., 327 (2014), 993-1009. doi: 10.1007/s00220-014-1942-z. |
[22] | K. Krupchyk and G. Uhlmann, Inverse boundary problems for polyharmonic operators with unbounded potentials, J. Spectr. Theory, 6 (2016), 145-183. doi: 10.4171/JST/122. |
[23] | A. I. Nachman, Inverse scattering at fixed energy, Mathematical physics, X (Leipzig, 1991), 434–441, Springer, Berlin, 1992. doi: 10.1007/978-3-642-77303-7_48. |
[24] | W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. |
[25] | G. Nakamura, Z. Sun and G. Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann., 303 (1995), 377-388. doi: 10.1007/BF01460996. |
[26] | R. G. Novikov, A multidimensional inverse spectral problem for the equation $-\Delta\psi+(v(x)-Eu(x))\psi = 0$, Funktsional. Anal. i Prilozhen, 22 (1988), 11-22. doi: 10.1007/BF01077418. |
[27] | L. Päivärinta, A. Panchenko and G. Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana, 19 (2003), 57-72. doi: 10.4171/RMI/338. |
[28] | G. de Rham, Differentiable Manifolds. Forms, Currents, Harmonic Forms, Grundlehren der Mathematischen Wissenschaften, 266. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-61752-2. |
[29] | T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411. |
[30] | M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67pp. |
[31] | M. Salo and L. Tzou, Carleman estimates and inverse problems for Dirac operators, Math. Ann., 344 (2009), 161-184. doi: 10.1007/s00208-008-0301-9. |
[32] | Z. Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc., 338 (1993), 953-969. doi: 10.2307/2154438. |
[33] | J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291. |
[34] | C. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian, SIAM J. Math. Anal., 29 (1998), 116-133. doi: 10.1137/S0036141096301038. |
[35] | H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Heidelberg: Johann Ambrosius Barth, 1995. |
[36] | G. Tuhin, An inverse problem on determining up to first order perturbations of a fourth order operator with partial boundary data, Inverse Problems, 31 (2015), 105009, 19pp. doi: 10.1088/0266-5611/31/10/105009. |
[37] | G. Tuhin and V. P. Krishnan, Determination of lower order perturbations of the polyharmonic operator from partial boundary data., Appl. Anal., 95 (2016), 2444-2463. doi: 10.1080/00036811.2015.1092522. |