Local uniqueness for an inverse boundary value problem with partial data

In dimension $n\geq 3$, we prove a local uniqueness result for the potentials $q$ of the Schr\"odinger equation $-\Delta u+qu=0$ from partial boundary data. More precisely, we show that potentials $q_1,q_2\in L^\infty$ with positive essential infima can be distinguished by local boundary data if there is a neighborhood of a boundary part where $q_1\geq q_2$ and $q_1\not\equiv q_2$.

[1]  Adrian Nachman,et al.  Reconstruction in the Calderón Problem with Partial Data , 2009 .

[2]  C. Kenig,et al.  The Calderón problem with partial data on manifolds and applications , 2012, 1211.1054.

[3]  A. Bukhgeǐm,et al.  Recovering a potential from Cauchy data in the two-dimensional case , 2008 .

[4]  Mikko Salo,et al.  Recent progress in the Calderon problem with partial data , 2013, 1302.4218.

[5]  Masaru Ikehata,et al.  Size estimation of inclusion , 1998 .

[6]  R. Regbaoui.  Unique Continuation from Sets of Positive Measure , 2001 .

[7]  G. Uhlmann,et al.  The Calderón problem with partial data , 2004, math/0405486.

[8]  Robert V. Kohn,et al.  Determining conductivity by boundary measurements , 1984 .

[9]  Masahiro Yamamoto,et al.  The Calderón problem with partial data in two dimensions , 2010 .

[10]  V. Isakov.  On uniqueness in the inverse conductivity problem with local data , 2007 .

[11]  Bastian von Harrach,et al.  Monotonicity-Based Shape Reconstruction in Electrical Impedance Tomography , 2013, SIAM J. Math. Anal..

[12]  Kari Astala,et al.  Calderon's inverse conductivity problem in the plane , 2006 .

[13]  Jean-Pierre Gossez,et al.  Strict Monotonicity of Eigenvalues and Unique Continuation , 1992 .

[14]  J. Sylvester,et al.  A global uniqueness theorem for an inverse boundary value problem , 1987 .

[15]  Boaz Haberman,et al.  Uniqueness in Calderón’s problem with Lipschitz conductivities , 2011, 1108.6068.

[16]  N. Tsouli,et al.  Strong unique continuation of eigenfunctions for p-Laplacian operator , 2001 .

[17]  L. Hörmander.  The Analysis of Linear Partial Differential Operators III , 2007 .

[18]  R. Kohn,et al.  Determining conductivity by boundary measurements II. Interior results , 1985 .

[19]  A. Calderón,et al.  On an inverse boundary value problem , 2006 .

[20]  G. Uhlmann,et al.  The Calderón problem with partial data , 2004 .

[21]  B. Harrach.  On uniqueness in diffuse optical tomography , 2009 .

[22]  Bastian Gebauer,et al.  Localized potentials in electrical impedance tomography , 2008 .

[23]  Bastian Harrach,et al.  Simultaneous determination of the diffusion and absorption coefficient from boundary data , 2012 .

[24]  Jin Keun Seo,et al.  The inverse conductivity problem with one measurement: stability and estimation of size , 1997 .

[25]  Otmar Scherzer,et al.  Detecting Interfaces in a Parabolic-Elliptic Problem from Surface Measurements , 2007, SIAM J. Numer. Anal..

[26]  A. Nachman,et al.  Global uniqueness for a two-dimensional inverse boundary value problem , 1996 .

[27]  N. Bourbaki.  Topological Vector Spaces , 1987 .