Inverse boundary value problems for the perturbed polyharmonic operator

Krupchyk, Katsiaryna; Lassas, Matti; Uhlmann, Gunther

doi:10.1090/S0002-9947-2013-05713-3

Inverse boundary value problems for the perturbed polyharmonic operator
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by Katsiaryna Krupchyk, Matti Lassas and Gunther Uhlmann PDF

Trans. Amer. Math. Soc. 366 (2014), 95-112 Request permission

Abstract:

We show that a first order perturbation $A(x)\cdot D+q(x)$ of the polyharmonic operator $(-\Delta )^m$, $m\ge 2$, can be determined uniquely from the set of the Cauchy data for the perturbed polyharmonic operator on a bounded domain in $\mathbb {R}^n$, $n\ge 3$. Notice that the corresponding result does not hold in general when $m=1$.

References

Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
Kari Astala and Lassi Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), no. 1, 265–299. MR 2195135, DOI 10.4007/annals.2006.163.265
Kari Astala, Lassi Päivärinta, and Matti Lassas, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations 30 (2005), no. 1-3, 207–224. MR 2131051, DOI 10.1081/PDE-200044485
A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), no. 1, 19–33. MR 2387648, DOI 10.1515/jiip.2008.002
Alexander L. Bukhgeim and Gunther Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations 27 (2002), no. 3-4, 653–668. MR 1900557, DOI 10.1081/PDE-120002868
Alberto P. Calderón, On an inverse boundary value problem, Comput. Appl. Math. 25 (2006), no. 2-3, 133–138. MR 2321646, DOI 10.1590/S0101-82052006000200002
Jin Cheng, Gen Nakamura, and Erkki Somersalo, Uniqueness of identifying the convection term, Commun. Korean Math. Soc. 16 (2001), no. 3, 405–413. Second Japan-Korea Joint Seminar on Inverse Problems and Related Topics (Seoul, 2001). MR 1843524
David Dos Santos Ferreira, Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, Determining a magnetic Schrödinger operator from partial Cauchy data, Comm. Math. Phys. 271 (2007), no. 2, 467–488. MR 2287913, DOI 10.1007/s00220-006-0151-9
G. Eskin, Global uniqueness in the inverse scattering problem for the Schrödinger operator with external Yang-Mills potentials, Comm. Math. Phys. 222 (2001), no. 3, 503–531. MR 1888087, DOI 10.1007/s002200100522
G. Eskin and J. Ralston, Inverse scattering problem for the Schrödinger equation with magnetic potential at a fixed energy, Comm. Math. Phys. 173 (1995), no. 1, 199–224. MR 1355624, DOI 10.1007/BF02100187
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, On nonuniqueness for Calderón’s inverse problem, Math. Res. Lett. 10 (2003), no. 5-6, 685–693. MR 2024725, DOI 10.4310/MRL.2003.v10.n5.a11
Allan Greenleaf, Matti Lassas, and Gunther Uhlmann, The Calderón problem for conormal potentials. I. Global uniqueness and reconstruction, Comm. Pure Appl. Math. 56 (2003), no. 3, 328–352. MR 1941812, DOI 10.1002/cpa.10061
Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys. 275 (2007), no. 3, 749–789. MR 2336363, DOI 10.1007/s00220-007-0311-6
Allan Greenleaf, Yaroslav Kurylev, Matti Lassas, and Gunther Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 55–97. MR 2457072, DOI 10.1090/S0273-0979-08-01232-9
Greenleaf, A., Kurylev, Y., Lassas, M., and Uhlmann, G., Approximate quantum and acoustic cloaking, to appear in Journal of Spectral Theory.
Gerd Grubb, Distributions and operators, Graduate Texts in Mathematics, vol. 252, Springer, New York, 2009. MR 2453959
Guillarmou, C., Tzou, L., Identification of a connection from Cauchy data space on a Riemann surface with boundary, to appear in GAFA.
Guillarmou, C., Tzou, L., Calderón inverse problem with partial data on Riemann surfaces, to appear in Duke Math. J.
Lars Hörmander, The analysis of linear partial differential operators. I, Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis; Reprint of the second (1990) edition [Springer, Berlin; MR1065993 (91m:35001a)]. MR 1996773, DOI 10.1007/978-3-642-61497-2
Masaru Ikehata, A special Green’s function for the biharmonic operator and its application to an inverse boundary value problem, Comput. Math. Appl. 22 (1991), no. 4-5, 53–66. Multidimensional inverse problems. MR 1127214, DOI 10.1016/0898-1221(91)90131-M
Oleg Yu. Imanuvilov, Gunther Uhlmann, and Masahiro Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc. 23 (2010), no. 3, 655–691. MR 2629983, DOI 10.1090/S0894-0347-10-00656-9
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
A. Katchalov, Y. Kurylev, M. Lassas, and N. Mandache, Equivalence of time-domain inverse problems and boundary spectral problems, Inverse Problems 20 (2004), no. 2, 419–436. MR 2065431, DOI 10.1088/0266-5611/20/2/007
Carlos E. Kenig, Johannes Sjöstrand, and Gunther Uhlmann, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), no. 2, 567–591. MR 2299741, DOI 10.4007/annals.2007.165.567
Kim Knudsen and Mikko Salo, Determining nonsmooth first order terms from partial boundary measurements, Inverse Probl. Imaging 1 (2007), no. 2, 349–369. MR 2282273, DOI 10.3934/ipi.2007.1.349
R. V. Kohn, H. Shen, M. S. Vogelius, and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography, Inverse Problems 24 (2008), no. 1, 015016, 21. MR 2384775, DOI 10.1088/0266-5611/24/1/015016
Matti Lassas, Michael Taylor, and Gunther Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), no. 2, 207–221. MR 2014876, DOI 10.4310/CAG.2003.v11.n2.a2
Matti Lassas and Gunther Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 771–787 (English, with English and French summaries). MR 1862026, DOI 10.1016/S0012-9593(01)01076-X
John M. Lee and Gunther Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), no. 8, 1097–1112. MR 1029119, DOI 10.1002/cpa.3160420804
Ulf Leonhardt, Optical conformal mapping, Science 312 (2006), no. 5781, 1777–1780. MR 2237569, DOI 10.1126/science.1126493
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, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), no. 1, 71–96. MR 1370758, DOI 10.2307/2118653
Gen Nakamura, Zi Qi Sun, and Gunther Uhlmann, Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field, Math. Ann. 303 (1995), no. 3, 377–388. MR 1354996, DOI 10.1007/BF01460996
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), no. 4, 11–22, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 4, 263–272 (1989). MR 976992, DOI 10.1007/BF01077418
J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312 (2006), no. 5781, 1780–1782. MR 2237570, DOI 10.1126/science.1125907
Lassi Päivärinta, Alexander Panchenko, and Gunther Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana 19 (2003), no. 1, 57–72. MR 1993415, DOI 10.4171/RMI/338
Mikko Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss. 139 (2004), 67. Dissertation, University of Helsinki, Helsinki, 2004. MR 2105191
Zi Qi Sun, An inverse boundary value problem for Schrödinger operators with vector potentials, Trans. Amer. Math. Soc. 338 (1993), no. 2, 953–969. MR 1179400, DOI 10.1090/S0002-9947-1993-1179400-1
John Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), no. 2, 201–232. MR 1038142, DOI 10.1002/cpa.3160430203
John Sylvester and Gunther Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), no. 1, 153–169. MR 873380, DOI 10.2307/1971291
Carlos F. Tolmasky, Exponentially growing solutions for nonsmooth first-order perturbations of the Laplacian, SIAM J. Math. Anal. 29 (1998), no. 1, 116–133. MR 1617178, DOI 10.1137/S0036141096301038