Geodesic
Double logarithmic stability in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 78-94 Cet article a éte moissonné depuis la source Math-Net.Ru

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We examine the stability issue in the inverse problem of determining a scalar potential appearing in the stationary Schrödinger equation in a bounded domain, from a partial elliptic Dirichlet-to-Neumann map. Namely, the Dirichlet data is imposed on the shadowed face of the boundary of the domain and the Neumann data is measured on its illuminated face. We establish a $\log\log$ stability estimate for the $L^2$-norm (resp. the $H^{-1}$-norm) of $H^t$, for $t>0$, and bounded (resp. $L^2$) potentials.
Keywords: inverse problem; stability; Schrödinger equation. 
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     title = {Double logarithmic stability in the identification of a scalar potential by~a~partial elliptic {Dirichlet-to-Neumann} map},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
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M. Choulli; Y. Kian; E. Soccorsi. Double logarithmic stability in the identification of a scalar potential by a partial elliptic Dirichlet-to-Neumann map. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 8 (2015) no. 3, pp. 78-94. http://geodesic.mathdoc.fr/item/VYURU_2015_8_3_a4/

[1] Lions J.-L., Magenes E., Non-Homogenuous Boundary Value Problems and Applications, v. I, Springer, Berlin, 1972

[2] Calderón A., “On an Inverse Boundary Problem”, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasileira de Matemática, Rio de Janeíro, 1980, 65–73 | MR

[3] Sylvester J., Uhlmann G., “A Global Uniqueness Theorem for an Inverse Boundary Value Problem”, Ann. of Math., 125 (1987), 153–169 | DOI | MR | Zbl

[4] Alessandrini G., “Stable Determination of Conductivity by Boundary Measurements”, Appl. Anal., 27 (1988), 153–172 | DOI | MR | Zbl

[5] Mandache N., “Exponential Instability in an Inverse Problem for the Schrodinger Equation”, Inverse Problems, 17 (2001), 1435–1444 | DOI | MR | Zbl

[6] Bukhgeim A. L., Uhlmann G., “Recovering a Potential from Partial Cauchy Data”, Commun. Part. Diff. Equat., 27 (2002), 653–658 | DOI | MR

[7] Kenig C. E., Sjöstrand J., Uhlmann G., “The Calderòn Problem with Partial Data”, Ann. of Math., 165 (2007), 567–791 | DOI | MR

[8] Nachman A., Street B., “Reconstruction in the Calderòn Problem with Partial Data”, Commun. Part. Diff. Equat., 35 (2010), 375–390 | DOI | MR | Zbl

[9] Heck H., Wang J.-N., “Stability Estimate for the Inverse Boundary Value Problem by Partial Cauchy Data”, Inv. Probl., 22 (2006), 1787–1797 | DOI | MR

[10] Caro P., Dos Santos Ferreira D., Ruiz A., Stability Estimates for the Calderón Problem with Partial Data, arXiv: 1405.1217

[11] Caro P., Dos Santos Ferreira D., Ruiz A., Stability Estimates for the Radon Transform with Restricted Data and Applications, arXiv: 1211.1887v2

[12] Apraiz J., Escauriaza L., Wang G., Zhang C., Observability Inequalities and Measurable Sets, arXiv: 1002.4876 | MR

[13] Alessandrini G., Gaburro R., “The Local Calderon Problem and the Determination at the Boundary of the Conductivity”, Commun. Partial Differ. Equat., 34:8 (2009), 918–936 | DOI | MR | Zbl

[14] Dautray R., Lions J.-L., Mathematical Analysis and Numerical Methods for Science and Technology, v. II, Springer-Verlag, Berlin, 1988 | DOI | MR | Zbl