August  2022, 16(4): 943-966. doi: 10.3934/ipi.2022006

Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA

*Corresponding author: L.D. Gauthier

Received  August 2021 Revised  December 2021 Published  August 2022 Early access  February 2022

Fund Project: R.M. Brown is partially supported by a grant from the Simons Foundation (#422756)

We consider inverse boundary value problems for polyharmonic operators and in particular, the problem of recovering the coefficients of terms up to order one. The main interest of our result is that it further relaxes the regularity required to establish uniqueness. The proof relies on an averaging technique introduced by Haberman and Tataru for the study of an inverse boundary value problem for a second order operator.

Citation: R.M. Brown, L.D. Gauthier. Inverse boundary value problems for polyharmonic operators with non-smooth coefficients. Inverse Problems and Imaging, 2022, 16 (4) : 943-966. doi: 10.3934/ipi.2022006
References:
[1]

Y. M. Assylbekov, Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order, Inverse Problems, 32 (2016), 105009, 22 pp. doi: 10.1088/0266-5611/32/10/105009.

[2]

Y. M. Assylbekov, Corrigendum: Inverse problems for the perturbed polyharmonic operator with coefficients in {S}obolev spaces with non-positive order, Inverse Problems, 33 (2017), 099501, 22 pp, URL https://doi.org/10.1088/1361-6420/aa7b93. doi: 10.1088/0266-5611/32/10/105009.

[3]

Y. Assylbekov and K. Iyer, Determining rough first order perturbations of the polyharmonic operator, Inverse Probl. Imaging, 13 (2019), 1045-1066.  doi: 10.3934/ipi.2019047.

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[5]

J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, New York, 1976. doi: 10.1007/978-3-642-66451-9.

[6]

S. Bhattacharyya and T. Ghosh, Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator, J. Fourier Anal. Appl., 25 (2019), 661-683.  doi: 10.1007/s00041-018-9625-3.

[7]

S. Bhattacharyya and T. Ghosh, An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator, Math. Annalen, 2021. doi: 10.1007/s00208-021-02276-6.

[8]

M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators div and grad, In Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, vol. 1980 of Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 149 (1980), 5–40.

[9]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[10]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[11]

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.

[12]

R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^ p, p>2n$, J. Fourier Anal. Appl., 9 (2003), 563-574.  doi: 10.1007/s00041-003-0902-3.

[13]

A. P. Calderón, On an inverse boundary value problem, In Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeiro, (1980), 65–73.

[14]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum Math. Pi, 4 (2016), e2, 28 pp. doi: 10.1017/fmp.2015.9.

[15]

A. P. Choudhury and V. P. Krishnan, Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials, J. Math. Anal. Appl., 431 (2015), 300-316.  doi: 10.1016/j.jmaa.2015.05.054.

[16]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics, 1991. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[17]

T. Ghosh 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.

[18]

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.

[19]

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.

[20]

S. Ham, Y. Kwon and S. Lee, Uniqueness in the Calderón problem and bilinear restriction estimates, J. Funct. Anal., 281 (2021), Paper No. 109119, 58 pp. doi: 10.1016/j.jfa.2021.109119.

[21]

K. KrupchykM. Lassas and G. Uhlmann, Determining a first order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012), 1781-1801.  doi: 10.1016/j.jfa.2011.11.021.

[22]

K. KrupchykM. Lassas and G. Uhlmann, Inverse boundary value problems for the perturbed polyharmonic operator, Trans. Amer. Math. Soc., 366 (2014), 95-112.  doi: 10.1090/S0002-9947-2013-05713-3.

[23]

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.

[24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[25]

D. MitreaM. Mitrea and S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), 1295-1333.  doi: 10.3934/cpaa.2008.A.1295.

[26]

L. PäivärintaA. Panchenko and G. Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana, 19 (2003), 57-72.  doi: 10.4171/RMI/338.

[27]

F. Ponce-Vanegas, The bilinear strategy for Calderón's problem, Rev. Mat. Iberoam., 37 (2021), 2119-2160.  doi: 10.4171/rmi/1257.

[28]

F. Ponce-Vanegas, Reconstruction of the derivative of the conductivity at the boundary, Inverse Probl. Imaging, 14 (2020), 701-718.  doi: 10.3934/ipi.2020032.

[29]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.

[30]

V. S. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator, Inverse Problems, 32 (2016), 045002, 19 pp. doi: 10.1088/0266-5611/32/4/045002.

[31]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[32]

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation on Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611.  doi: 10.1016/0022-1236(84)90066-1.

[33]

G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J., 39 (1990), 671-702.  doi: 10.1512/iumj.1990.39.39034.

[34]

L. Yan, Inverse boundary problems for biharmonic operators in transversally anisotropic geometries, SIAM J. Math. Anal., 53 (2021), 6617-6653.  doi: 10.1137/21M1391419.

[35]

L. Yan, Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds, arXiv: 2109.07712.

show all references

References:
[1]

Y. M. Assylbekov, Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order, Inverse Problems, 32 (2016), 105009, 22 pp. doi: 10.1088/0266-5611/32/10/105009.

[2]

Y. M. Assylbekov, Corrigendum: Inverse problems for the perturbed polyharmonic operator with coefficients in {S}obolev spaces with non-positive order, Inverse Problems, 33 (2017), 099501, 22 pp, URL https://doi.org/10.1088/1361-6420/aa7b93. doi: 10.1088/0266-5611/32/10/105009.

[3]

Y. Assylbekov and K. Iyer, Determining rough first order perturbations of the polyharmonic operator, Inverse Probl. Imaging, 13 (2019), 1045-1066.  doi: 10.3934/ipi.2019047.

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math., 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[5]

J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, Berlin, New York, 1976. doi: 10.1007/978-3-642-66451-9.

[6]

S. Bhattacharyya and T. Ghosh, Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator, J. Fourier Anal. Appl., 25 (2019), 661-683.  doi: 10.1007/s00041-018-9625-3.

[7]

S. Bhattacharyya and T. Ghosh, An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator, Math. Annalen, 2021. doi: 10.1007/s00208-021-02276-6.

[8]

M. E. Bogovskiĭ, Solutions of some problems of vector analysis, associated with the operators div and grad, In Theory of Cubature Formulas and the Application of Functional Analysis to Problems of Mathematical Physics, vol. 1980 of Trudy Sem. S. L. Soboleva, No. 1, Akad. Nauk SSSR Sibirsk. Otdel. Inst. Mat., Novosibirsk, 149 (1980), 5–40.

[9]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations, Geom. Funct. Anal., 3 (1993), 107-156.  doi: 10.1007/BF01896020.

[10]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. II. The KdV-equation, Geom. Funct. Anal., 3 (1993), 209-262.  doi: 10.1007/BF01895688.

[11]

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.

[12]

R. Brown and R. Torres, Uniqueness in the inverse conductivity problem for conductivities with $3/2$ derivatives in $L^ p, p>2n$, J. Fourier Anal. Appl., 9 (2003), 563-574.  doi: 10.1007/s00041-003-0902-3.

[13]

A. P. Calderón, On an inverse boundary value problem, In Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeiro, (1980), 65–73.

[14]

P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum Math. Pi, 4 (2016), e2, 28 pp. doi: 10.1017/fmp.2015.9.

[15]

A. P. Choudhury and V. P. Krishnan, Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials, J. Math. Anal. Appl., 431 (2015), 300-316.  doi: 10.1016/j.jmaa.2015.05.054.

[16]

F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Lecture Notes in Mathematics, 1991. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12245-3.

[17]

T. Ghosh 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.

[18]

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.

[19]

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.

[20]

S. Ham, Y. Kwon and S. Lee, Uniqueness in the Calderón problem and bilinear restriction estimates, J. Funct. Anal., 281 (2021), Paper No. 109119, 58 pp. doi: 10.1016/j.jfa.2021.109119.

[21]

K. KrupchykM. Lassas and G. Uhlmann, Determining a first order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012), 1781-1801.  doi: 10.1016/j.jfa.2011.11.021.

[22]

K. KrupchykM. Lassas and G. Uhlmann, Inverse boundary value problems for the perturbed polyharmonic operator, Trans. Amer. Math. Soc., 366 (2014), 95-112.  doi: 10.1090/S0002-9947-2013-05713-3.

[23]

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.

[24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[25]

D. MitreaM. Mitrea and S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), 1295-1333.  doi: 10.3934/cpaa.2008.A.1295.

[26]

L. PäivärintaA. Panchenko and G. Uhlmann, Complex geometrical optics solutions for Lipschitz conductivities, Rev. Mat. Iberoamericana, 19 (2003), 57-72.  doi: 10.4171/RMI/338.

[27]

F. Ponce-Vanegas, The bilinear strategy for Calderón's problem, Rev. Mat. Iberoam., 37 (2021), 2119-2160.  doi: 10.4171/rmi/1257.

[28]

F. Ponce-Vanegas, Reconstruction of the derivative of the conductivity at the boundary, Inverse Probl. Imaging, 14 (2020), 701-718.  doi: 10.3934/ipi.2020032.

[29]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.

[30]

V. S. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator, Inverse Problems, 32 (2016), 045002, 19 pp. doi: 10.1088/0266-5611/32/4/045002.

[31]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[32]

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation on Lipschitz domains, J. Funct. Anal., 59 (1984), 572-611.  doi: 10.1016/0022-1236(84)90066-1.

[33]

G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Indiana Univ. Math. J., 39 (1990), 671-702.  doi: 10.1512/iumj.1990.39.39034.

[34]

L. Yan, Inverse boundary problems for biharmonic operators in transversally anisotropic geometries, SIAM J. Math. Anal., 53 (2021), 6617-6653.  doi: 10.1137/21M1391419.

[35]

L. Yan, Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds, arXiv: 2109.07712.

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