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

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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

Abstract
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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:

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[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
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[17]	M. Santacesaria, New global stability estimates for the Calderón problem in two dimensions,, J. Inst. Math. Jussieu, 12 (2013), 553. 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). doi: 10.1088/0266-5611/25/12/123011. Google Scholar
[20]	I. N. Vekua, Generalized Analytic Functions,, Pergamon Press, (1962). Google Scholar

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References:

[1]	R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, $2^{nd}$ edition, (2003). Google Scholar
[2]	G. Alessandrini, Stable determination of conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153. 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, (2010). Google Scholar
[5]	E. Blåsten, On the Gel'fand-Calderón Inverse Problem in Two Dimensions,, Ph.D. thesis, (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. doi: 10.1515/jiip.2008.002. Google Scholar
[7]	L. C. Evans, Partial Differential Equations,, American Mathematical Society, (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, (1972). Google Scholar
[11]	L. Liu, Stability Estimates for the Two Dimensional Inverse Conductivity Problem,, Ph.D. thesis, (1997). Google Scholar
[12]	N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation,, Inverse Problems, 17 (2001), 1435. doi: 10.1088/0266-5611/17/5/313. Google Scholar
[13]	C. Miranda, Partial Differential Equations of Elliptic Type,, $2^{nd}$ revised edition, (1970). Google Scholar
[14]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem,, Ann. of Math., 143 (1996), 71. 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. 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). doi: 10.1088/0266-5611/25/12/123011. Google Scholar
[20]	I. N. Vekua, Generalized Analytic Functions,, Pergamon Press, (1962). Google Scholar

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