Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials

Previous Article
The perturbation of transmission eigenvalues for inhomogeneous media in the presence of small penetrable inclusions
IPI Home
This Issue
Next Article
Periodic spline-based frames for image restoration

August 2015, 9(3): 709-723. doi: 10.3934/ipi.2015.9.709

Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials

Eemeli Blåsten ^1, , Oleg Yu. Imanuvilov ^2, and Masahiro Yamamoto ^3,

1.	Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, 19086 Tallinn, Estonia
2.	Department of Mathematics, Colorado State University,101 Weber Building, Fort Colins, CO 80523-1784, United States
3.	Department of Mathematical Sciences, The University of Tokyo, Komaba Meguro Tokyo 153-8914

Received October 2014 Revised February 2015 Published July 2015

Abstract
Related Papers

We consider inverse boundary value problems for the Schrödinger equations in two dimensions. Within less regular classes of potentials, we establish a conditional stability estimate of logarithmic order. Moreover we prove the uniqueness within $L^p$-class of potentials with $p>2$.

Keywords: stability, uniqueness., Calderón problem.

Mathematics Subject Classification: Primary: 35R30, 30G20; Secondary: 35J1.

Citation: Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems and Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709

References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.
[2]	G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.
[3]	K. Astala, D. Faraco and K. M. Rogers, Rough potential recovery in the plane, preprint,, , ().
[4]	E. Blåsten, The Inverse Problem of the Schrödinger Equation in the Plane: A Dissection of Bukhgeim's Result, Licentiate thesis, University of Helsinki, 2010.
[5]	E. Blåsten, On the Gel'fand-Calderón Inverse Problem in Two Dimensions, Ph.D. thesis, University of Helsinki, 2013.
[6]	A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.
[7]	L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
[8]	O. Yu. Imanuvilov and M. Yamamoto, Inverse boundary value problem for linear Schrödinger equation in two dimensions, preprint,, , ().
[9]	O. Yu. Imanuvilov and M. Yamamoto, Uniqueness for inverse boundary value problems, by Dirichlet-to-Neumann map on subboundaries, (). doi: 10.1007/s00032-013-0205-3.
[10]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
[11]	L. Liu, Stability Estimates for the Two Dimensional Inverse Conductivity Problem, Ph.D. thesis, University of Rochester, 1997.
[12]	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.
[13]	C. Miranda, Partial Differential Equations of Elliptic Type, $2^{nd}$ revised edition, Springer-Verlag, New York-Berlin, 1970.
[14]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.
[15]	R. G. Novikov and M. Santacesaria, A global stability estimate for the Gel'fand-Calderón, inverse problem in two dimensions, (). doi: 10.1515/JIIP.2011.003.
[16]	R. G. Novikov and M. Santacesaria, Global uniqueness and reconstruction for the, multi-channel Gel'fand-Calderón inverse problem in two dimensions., (). doi: 10.1016/j.bulsci.2011.04.007.
[17]	M. Santacesaria, New global stability estimates for the Calderón problem in two dimensions, J. Inst. Math. Jussieu, 12 (2013), 553-569. doi: 10.1017/S147474801200076X.
[18]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, problem., (). doi: 10.2307/1971291.
[19]	G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.
[20]	I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London-Paris-Frankfurt, 1962.

show all references

References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.
[2]	G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172. doi: 10.1080/00036818808839730.
[3]	K. Astala, D. Faraco and K. M. Rogers, Rough potential recovery in the plane, preprint,, , ().
[4]	E. Blåsten, The Inverse Problem of the Schrödinger Equation in the Plane: A Dissection of Bukhgeim's Result, Licentiate thesis, University of Helsinki, 2010.
[5]	E. Blåsten, On the Gel'fand-Calderón Inverse Problem in Two Dimensions, Ph.D. thesis, University of Helsinki, 2013.
[6]	A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl., 16 (2008), 19-33. doi: 10.1515/jiip.2008.002.
[7]	L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
[8]	O. Yu. Imanuvilov and M. Yamamoto, Inverse boundary value problem for linear Schrödinger equation in two dimensions, preprint,, , ().
[9]	O. Yu. Imanuvilov and M. Yamamoto, Uniqueness for inverse boundary value problems, by Dirichlet-to-Neumann map on subboundaries, (). doi: 10.1007/s00032-013-0205-3.
[10]	J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.
[11]	L. Liu, Stability Estimates for the Two Dimensional Inverse Conductivity Problem, Ph.D. thesis, University of Rochester, 1997.
[12]	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.
[13]	C. Miranda, Partial Differential Equations of Elliptic Type, $2^{nd}$ revised edition, Springer-Verlag, New York-Berlin, 1970.
[14]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math., 143 (1996), 71-96. doi: 10.2307/2118653.
[15]	R. G. Novikov and M. Santacesaria, A global stability estimate for the Gel'fand-Calderón, inverse problem in two dimensions, (). doi: 10.1515/JIIP.2011.003.
[16]	R. G. Novikov and M. Santacesaria, Global uniqueness and reconstruction for the, multi-channel Gel'fand-Calderón inverse problem in two dimensions., (). doi: 10.1016/j.bulsci.2011.04.007.
[17]	M. Santacesaria, New global stability estimates for the Calderón problem in two dimensions, J. Inst. Math. Jussieu, 12 (2013), 553-569. doi: 10.1017/S147474801200076X.
[18]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, problem., (). doi: 10.2307/1971291.
[19]	G. Uhlmann, Electrical impedance tomography and Calderón's problem, Inverse Problems, 25 (2009), 123011, 39 pp. doi: 10.1088/0266-5611/25/12/123011.
[20]	I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London-Paris-Frankfurt, 1962.

[1]	Pedro Caro, Mikko Salo. Stability of the Calderón problem in admissible geometries. Inverse Problems and Imaging, 2014, 8 (4) : 939-957. doi: 10.3934/ipi.2014.8.939
[2]	Albert Clop, Daniel Faraco, Alberto Ruiz. Stability of Calderón's inverse conductivity problem in the plane for discontinuous conductivities. Inverse Problems and Imaging, 2010, 4 (1) : 49-91. doi: 10.3934/ipi.2010.4.49
[3]	Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
[4]	Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems and Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046
[5]	María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049
[6]	Matteo Santacesaria. Note on Calderón's inverse problem for measurable conductivities. Inverse Problems and Imaging, 2019, 13 (1) : 149-157. doi: 10.3934/ipi.2019008
[7]	Oleg Yu. Imanuvilov, Masahiro Yamamoto. Calderón problem for Maxwell's equations in cylindrical domain. Inverse Problems and Imaging, 2014, 8 (4) : 1117-1137. doi: 10.3934/ipi.2014.8.1117
[8]	Yernat M. Assylbekov. Reconstruction in the partial data Calderón problem on admissible manifolds. Inverse Problems and Imaging, 2017, 11 (3) : 455-476. doi: 10.3934/ipi.2017021
[9]	Kim Knudsen, Aksel Kaastrup Rasmussen. Direct regularized reconstruction for the three-dimensional Calderón problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022002
[10]	Henrik Garde, Nuutti Hyvönen. Reconstruction of singular and degenerate inclusions in Calderón's problem. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022021
[11]	Fabrice Delbary, Kim Knudsen. Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem. Inverse Problems and Imaging, 2014, 8 (4) : 991-1012. doi: 10.3934/ipi.2014.8.991
[12]	Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297
[13]	Sun-Sig Byun, Yunsoo Jang. Calderón-Zygmund estimate for homogenization of parabolic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6689-6714. doi: 10.3934/dcds.2016091
[14]	Marius Ionescu, Luke G. Rogers. Complex Powers of the Laplacian on Affine Nested Fractals as Calderón-Zygmund operators. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2155-2175. doi: 10.3934/cpaa.2014.13.2155
[15]	Sun-Sig Byun, Yumi Cho, Shuang Liang. Calderón-Zygmund estimates for quasilinear elliptic double obstacle problems with variable exponent and logarithmic growth. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3843-3855. doi: 10.3934/dcdsb.2020038
[16]	Kim Knudsen, Jennifer L. Mueller. The born approximation and Calderón's method for reconstruction of conductivities in 3-D. Conference Publications, 2011, 2011 (Special) : 844-853. doi: 10.3934/proc.2011.2011.844
[17]	Naoki Shioji, Kohtaro Watanabe. Uniqueness of positive radial solutions of the Brezis-Nirenberg problem on thin annular domains on $ {\mathbb S}^n $ and symmetry breaking bifurcations. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4727-4770. doi: 10.3934/cpaa.2020210
[18]	Ali Wehbe, Rayan Nasser, Nahla Noun. Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions. Mathematical Control and Related Fields, 2021, 11 (4) : 885-904. doi: 10.3934/mcrf.2020050
[19]	H. Gajewski, I. V. Skrypnik. To the uniqueness problem for nonlinear parabolic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 315-336. doi: 10.3934/dcds.2004.10.315
[20]	Xin Lai, Xinfu Chen, Mingxin Wang, Cong Qin, Yajing Zhang. Existence, uniqueness, and stability of bubble solutions of a chemotaxis model. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 805-832. doi: 10.3934/dcds.2016.36.805

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Eemeli Blåsten Oleg Yu. Imanuvilov Masahiro Yamamoto
on Google Scholar
Eemeli Blåsten Oleg Yu. Imanuvilov Masahiro Yamamoto

[Back to Top]