Determining rough first order perturbations of the polyharmonic operator

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

References:

[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.

show all references

References:

[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.

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