American Institute of Mathematical Sciences

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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

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 & 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.  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]

K. Astala, D. Faraco and K. M. Rogers, Rough potential recovery in the plane, preprint,, , ().   Google Scholar

[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. Google Scholar

[5]

E. Blåsten, On the Gel'fand-Calderón Inverse Problem in Two Dimensions, Ph.D. thesis, University of Helsinki, 2013. Google Scholar

[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.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[8]

O. Yu. Imanuvilov and M. Yamamoto, Inverse boundary value problem for linear Schrödinger equation in two dimensions, preprint,, , ().   Google Scholar

[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.  Google Scholar

[10]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[11]

L. Liu, Stability Estimates for the Two Dimensional Inverse Conductivity Problem, Ph.D. thesis, University of Rochester, 1997.  Google Scholar

[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.  Google Scholar

[13]

C. Miranda, Partial Differential Equations of Elliptic Type, $2^{nd}$ revised edition, Springer-Verlag, New York-Berlin, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, problem., ().  doi: 10.2307/1971291.  Google Scholar

[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.  Google Scholar

[20]

I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London-Paris-Frankfurt, 1962.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, $2^{nd}$ edition, Elsevier/Academic Press, Amsterdam, 2003.  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]

K. Astala, D. Faraco and K. M. Rogers, Rough potential recovery in the plane, preprint,, , ().   Google Scholar

[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. Google Scholar

[5]

E. Blåsten, On the Gel'fand-Calderón Inverse Problem in Two Dimensions, Ph.D. thesis, University of Helsinki, 2013. Google Scholar

[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.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.  Google Scholar

[8]

O. Yu. Imanuvilov and M. Yamamoto, Inverse boundary value problem for linear Schrödinger equation in two dimensions, preprint,, , ().   Google Scholar

[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.  Google Scholar

[10]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[11]

L. Liu, Stability Estimates for the Two Dimensional Inverse Conductivity Problem, Ph.D. thesis, University of Rochester, 1997.  Google Scholar

[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.  Google Scholar

[13]

C. Miranda, Partial Differential Equations of Elliptic Type, $2^{nd}$ revised edition, Springer-Verlag, New York-Berlin, 1970.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value, problem., ().  doi: 10.2307/1971291.  Google Scholar

[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.  Google Scholar

[20]

I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London-Paris-Frankfurt, 1962.  Google Scholar

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