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October  2019, 13(5): 1045-1066. doi: 10.3934/ipi.2019047

Determining rough first order perturbations of the polyharmonic operator

Yernat Assylbekov 1, and  Karthik Iyer 2,

1. 

Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005, USA
2. 

The Vanguard Group, Malvern, PA 19335, USA

Received  November 2018 Published  July 2019

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.

Keywords: Polyharmonic, perturbations, uniqueness, inverse problems.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J62.
Citation: Yernat Assylbekov, Karthik Iyer. Determining rough first order perturbations of the polyharmonic operator. Inverse Problems & Imaging, 2019, 13 (5) : 1045-1066. doi: 10.3934/ipi.2019047
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[4]

A. Behzadan and N. Holst, Multiplication in Sobolev Spaces, Revisited, arXiv: 1512.07379. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035.   Google Scholar

[10]

G. Folland, Real Analysis, Modern Techniques and their Applications, John Wiley & Sons, New York, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, 252. Springer, New York, 2009.  Google Scholar

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B. Haberman, Unique determination of a magnetic Schrdinger operator with unbounded magnetic potential from boundary data, arXiv: 1512.01580. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Heidelberg: Johann Ambrosius Barth, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[4]

A. Behzadan and N. Holst, Multiplication in Sobolev Spaces, Revisited, arXiv: 1512.07379. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

L. D. Faddeev, Increasing solutions of the Schrödinger equation, Soviet Physics Doklady, 10 (1966), 1033-1035.   Google Scholar

[10]

G. Folland, Real Analysis, Modern Techniques and their Applications, John Wiley & Sons, New York, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

G. Grubb, Distributions and Operators, Graduate Texts in Mathematics, 252. Springer, New York, 2009.  Google Scholar

[14]

B. Haberman, Unique determination of a magnetic Schrdinger operator with unbounded magnetic potential from boundary data, arXiv: 1512.01580. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

M. Salo, Inverse problems for nonsmooth first order perturbations of the Laplacian, Ann. Acad. Sci. Fenn. Math. Diss., 139 (2004), 67pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Heidelberg: Johann Ambrosius Barth, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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