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October  2020, 14(5): 783-796. doi: 10.3934/ipi.2020036

Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

Department of Mathmatics, University of California, Irvine, Irvine, CA 92697, USA

*Corresponding author: Boya Liu

Received  November 2019 Revised  April 2020 Published  July 2020

Fund Project: The research is partially supported by the National Science Foundation (DMS 1815922)

We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability for this problem were established in [37] and [7], respectively. We establish stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials, sharpening those of [7].

Citation: Boya Liu. Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies. Inverse Problems & Imaging, 2020, 14 (5) : 783-796. doi: 10.3934/ipi.2020036
References:
[1]

M. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-14648-5.  Google Scholar

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[3]

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

[4]

Y. Assylbekov and Y. Yang, Determining the first order perturbation of a polyharmonic operator on admissible manifolds, J. Differential Equations, 262 (2017), 590-614.  doi: 10.1016/j.jde.2016.09.039.  Google Scholar

[5]

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.  Google Scholar

[6]

P. Caro and K. Marinov, Stability of inverse problems in an infinite slab with partial data, Comm. Partial Differential Equations, 41 (2016), 683-704.  doi: 10.1080/03605302.2015.1127967.  Google Scholar

[7]

A. Choudhury and H. Heck, Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates, J. Inverse Ill-Posed Probl., 25 (2017), 251-263.  doi: 10.1515/jiip-2016-0019.  Google Scholar

[8]

A. Choudhury and H. Heck, Increasing stability for the inverse problem for the Schrödinger equation, Math. Methods Appl. Sci., 41 (2018), 606-614.  doi: 10.1002/mma.4632.  Google Scholar

[9]

A. Choudhury and V. 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.  Google Scholar

[10]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[11]

D. Faraco and K. M. Rogers, The Sobolev norm of characteristic functions with applications to the Calderón inverse problem, Quart. J. Math., 64 (2013), 133-147.  doi: 10.1093/qmath/har039.  Google Scholar

[12]

J. Feldman, M. Salo and G. Uhlmann, The Calderón problem: An Introduction to Inverse Problems, Preliminary notes on the book in preparation, 2019. Google Scholar

[13]

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

[14]

T. Ghosh and V. 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

[15]

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

[16]

H. Heck and J. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 369-383.  doi: 10.13137/2464-8728/13164.  Google Scholar

[17]

P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308.  doi: 10.1006/jdeq.1996.0096.  Google Scholar

[18]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004.  Google Scholar

[19]

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

[20]

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

[21]

V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.  doi: 10.3934/ipi.2007.1.95.  Google Scholar

[22]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control Methods in PDE-dynamical systems, Contemp. Math., 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.  Google Scholar

[23]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.  Google Scholar

[24]

V. IsakovR. Lai and J. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594.  doi: 10.1137/15M1019052.  Google Scholar

[25]

V. IsakovS. NagayasuG. Uhlmann and J. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemp. Math., 615 (2014), 131-141.  doi: 10.1090/conm/615/12268.  Google Scholar

[26]

V. Isakov and J. Wang, Increasing stability for determining the potential in the Schr${\rm{\ddot d}}$inger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 8 (2014), 1139-1150.  doi: 10.3934/ipi.2014.8.1139.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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

[30]

K. Krupchyk and G. Uhlmann, Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.  doi: 10.1080/03605302.2018.1446163.  Google Scholar

[31]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.  doi: 10.1016/j.matpur.2019.02.017.  Google Scholar

[32]

L. Liang, Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data, Inverse Probl. Imaging, 9 (2015), 469-478.  doi: 10.3934/ipi.2015.9.469.  Google Scholar

[33]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.  Google Scholar

[34] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[35]

V. 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.  Google Scholar

[36]

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.  Google Scholar

[37]

Y. Yang, Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data, J. Differential Equations, 257 (2014), 3607-3639.  doi: 10.1016/j.jde.2014.07.003.  Google Scholar

show all references

References:
[1]

M. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics, Springer, Cham, 2015. doi: 10.1007/978-3-319-14648-5.  Google Scholar

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[3]

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

[4]

Y. Assylbekov and Y. Yang, Determining the first order perturbation of a polyharmonic operator on admissible manifolds, J. Differential Equations, 262 (2017), 590-614.  doi: 10.1016/j.jde.2016.09.039.  Google Scholar

[5]

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.  Google Scholar

[6]

P. Caro and K. Marinov, Stability of inverse problems in an infinite slab with partial data, Comm. Partial Differential Equations, 41 (2016), 683-704.  doi: 10.1080/03605302.2015.1127967.  Google Scholar

[7]

A. Choudhury and H. Heck, Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates, J. Inverse Ill-Posed Probl., 25 (2017), 251-263.  doi: 10.1515/jiip-2016-0019.  Google Scholar

[8]

A. Choudhury and H. Heck, Increasing stability for the inverse problem for the Schrödinger equation, Math. Methods Appl. Sci., 41 (2018), 606-614.  doi: 10.1002/mma.4632.  Google Scholar

[9]

A. Choudhury and V. 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.  Google Scholar

[10]

D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137. doi: 10.1088/0266-5611/19/6/057.  Google Scholar

[11]

D. Faraco and K. M. Rogers, The Sobolev norm of characteristic functions with applications to the Calderón inverse problem, Quart. J. Math., 64 (2013), 133-147.  doi: 10.1093/qmath/har039.  Google Scholar

[12]

J. Feldman, M. Salo and G. Uhlmann, The Calderón problem: An Introduction to Inverse Problems, Preliminary notes on the book in preparation, 2019. Google Scholar

[13]

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

[14]

T. Ghosh and V. 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

[15]

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

[16]

H. Heck and J. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 369-383.  doi: 10.13137/2464-8728/13164.  Google Scholar

[17]

P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308.  doi: 10.1006/jdeq.1996.0096.  Google Scholar

[18]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004.  Google Scholar

[19]

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

[20]

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

[21]

V. Isakov, On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.  doi: 10.3934/ipi.2007.1.95.  Google Scholar

[22]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control Methods in PDE-dynamical systems, Contemp. Math., 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.  Google Scholar

[23]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.  Google Scholar

[24]

V. IsakovR. Lai and J. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594.  doi: 10.1137/15M1019052.  Google Scholar

[25]

V. IsakovS. NagayasuG. Uhlmann and J. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemp. Math., 615 (2014), 131-141.  doi: 10.1090/conm/615/12268.  Google Scholar

[26]

V. Isakov and J. Wang, Increasing stability for determining the potential in the Schr${\rm{\ddot d}}$inger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 8 (2014), 1139-1150.  doi: 10.3934/ipi.2014.8.1139.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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

[30]

K. Krupchyk and G. Uhlmann, Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.  doi: 10.1080/03605302.2018.1446163.  Google Scholar

[31]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.  doi: 10.1016/j.matpur.2019.02.017.  Google Scholar

[32]

L. Liang, Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data, Inverse Probl. Imaging, 9 (2015), 469-478.  doi: 10.3934/ipi.2015.9.469.  Google Scholar

[33]

N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.  doi: 10.1088/0266-5611/17/5/313.  Google Scholar

[34] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.   Google Scholar
[35]

V. 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.  Google Scholar

[36]

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.  Google Scholar

[37]

Y. Yang, Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data, J. Differential Equations, 257 (2014), 3607-3639.  doi: 10.1016/j.jde.2014.07.003.  Google Scholar

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