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

Boya Liu

doi:10.3934/ipi.2020036

Article Contents

2020, Volume 14, Issue 5: 783-796. Doi: 10.3934/ipi.2020036

This issue Previous Article Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements Next Article Numerical recovery of magnetic diffusivity in a three dimensional spherical dynamo equation

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

Boya Liu^,

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

^*Corresponding author: Boya Liu
^*Corresponding author: Boya Liu

Received: November 2019

Revised: April 2020

Published: July 2020

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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].

Keywords:

Mathematics Subject Classification: Primary: 35R30, 35J40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.
[2]	G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[12]	J. Feldman, M. Salo and G. Uhlmann, The Calderón problem: An Introduction to Inverse Problems, Preliminary notes on the book in preparation, 2019.
[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.
[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.
[15]	G. Grubb, Distributions and Operators, Volume 252 of Graduate Texts in Mathematics. Springer, New York, 2009.
[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.
[17]	P. Hähner, A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308. doi: 10.1006/jdeq.1996.0096.
[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.
[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.
[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.
[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.
[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.
[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.
[24]	V. Isakov, R. Lai and J. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594. doi: 10.1137/15M1019052.
[25]	V. Isakov, S. Nagayasu, G. 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.
[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.
[27]	K. Krupchyk, M. 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.
[28]	K. Krupchyk, M. 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.
[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.
[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.
[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.
[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.
[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.
[34]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[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.
[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.
[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.