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May  2011, 5(2): 297-322. doi: 10.3934/ipi.2011.5.297

On an inverse problem in electromagnetism with local data: stability and uniqueness

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Received  May 2010 Revised  March 2011 Published  May 2011

In this paper we prove a stable determination of the coefficients of the time-harmonic Maxwell equations from local boundary data. The argument --due to Isakov-- requires some restrictions on the domain.
Citation: Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems & Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297
References:
[1]

G. Alessandrini, Stable determination of the conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153.  doi: 10.1080/00036818808839730.  Google Scholar

[2]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem,, Adv. Appl. Math., 35 (2005), 207.  doi: 10.1016/j.aam.2004.12.002.  Google Scholar

[3]

K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207.  doi: 10.1081/PDE-200044485.  Google Scholar

[4]

R. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities,, SIAM J. Math. Anal., 27 (1996), 1049.  doi: 10.1137/S0036141094271132.  Google Scholar

[5]

A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data,, Comm. PDE, 27 (2002), 653.  doi: 10.1081/PDE-120002868.  Google Scholar

[6]

P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data,, Comm. PDE, 34 (2009), 1425.  doi: 10.1080/03605300903296272.  Google Scholar

[7]

P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements,, Inverse Problems, 26 (2010).   Google Scholar

[8]

D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves,, Arch. Rational Mech. Anal., 119 (1992), 59.  doi: 10.1007/BF00376010.  Google Scholar

[9]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data,, Inverse Problems, 22 (2006), 1787.  doi: 10.1088/0266-5611/22/5/015.  Google Scholar

[10]

H. Heck and J.-N. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements,, preprint (2007) \arXiv{0708.3289v1}., (2007).   Google Scholar

[11]

V. Isakov, Carleman estimates and applications to inverse problems,, Milan J. Math., 72 (2004), 249.  doi: 10.1007/s00032-004-0033-6.  Google Scholar

[12]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95.   Google Scholar

[13]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Funct. Anal., 130 (1995), 161.  doi: 10.1006/jfan.1995.1067.  Google Scholar

[14]

M. Joshi, S. R. McDowall, Total determination of material parameters from electromagnetic boundary information,, Pacific J. Math., 193 (2000), 107.  doi: 10.2140/pjm.2000.193.107.  Google Scholar

[15]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math. J., 157 (2011), 369.  doi: 10.1215/00127094-1272903.  Google Scholar

[16]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data,, Ann. of Math., 165 (2007), 567.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[17]

Y. Kurylev, M. Lassas, Matti and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems,, J. Math. Pures Appl., 86 (2006), 237.  doi: 10.1016/j.matpur.2006.01.008.  Google Scholar

[18]

R. Leis, "Initial Boundary Value Problems in Mathematical Physics,", Wiley, (1986).   Google Scholar

[19]

H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering,, Inverse Problems, 23 (2007), 2357.  doi: 10.1088/0266-5611/23/6/005.  Google Scholar

[20]

S. R. McDowall, An electromagnetic inverse problem in chiral media,, Trans. Amer. Math. Soc., 352 (2000), 2993.  doi: 10.1090/S0002-9947-00-02518-6.  Google Scholar

[21]

M. Mitrea, Sharp Hodge decomposition, Maxwell's equations, and vector Poisson problems on non-smooth, three-dimensional riemannian manifolds,, Duke Math. J., 125 (2004), 467.  doi: 10.1215/S0012-7094-04-12322-1.  Google Scholar

[22]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke Math. J., 70 (1993), 617.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[23]

P. Ola, L. Päivärinta and E. Somersalo, Inverse problems for time harmonic electrodynamics. Inside out: inverse problems and applications,, 169-191, 47 (2003), 169.   Google Scholar

[24]

P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials,, SIAM J. Appl. Math., 56 (1996), 1129.  doi: 10.1137/S0036139995283948.  Google Scholar

[25]

M. Salo and L. Tzou, Carleman estimates and inverse problems for Dirac operators,, Math. Ann., 344 (2009), 161.  doi: 10.1007/s00208-008-0301-9.  Google Scholar

[26]

M. Salo and L. Tzou, Inverse problems with partial data for a Dirac system: A Carleman estimate approach,, Adv. Math., 225 (2010), 487.  doi: 10.1016/j.aim.2010.03.003.  Google Scholar

[27]

E. Somersalo, D. Isaacson and M. Cheney, A linearized inverse boundary value problem for Maxwell's equations,, J. Comp. Appl. Math., 42 (1992), 123.  doi: 10.1016/0377-0427(92)90167-V.  Google Scholar

[28]

E. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970).   Google Scholar

[29]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

[30]

H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers,, Rev. Mat. Complut., 15 (2002), 475.   Google Scholar

show all references

References:
[1]

G. Alessandrini, Stable determination of the conductivity by boundary measurements,, Appl. Anal., 27 (1988), 153.  doi: 10.1080/00036818808839730.  Google Scholar

[2]

G. Alessandrini and S. Vessella, Lipschitz stability for the inverse conductivity problem,, Adv. Appl. Math., 35 (2005), 207.  doi: 10.1016/j.aam.2004.12.002.  Google Scholar

[3]

K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane,, Comm. PDE, 30 (2005), 207.  doi: 10.1081/PDE-200044485.  Google Scholar

[4]

R. Brown, Global uniqueness in the impedance imaging problem for less regular conductivities,, SIAM J. Math. Anal., 27 (1996), 1049.  doi: 10.1137/S0036141094271132.  Google Scholar

[5]

A. Bukhgeim and G. Uhlmann, Recovering a potential from partial Cauchy data,, Comm. PDE, 27 (2002), 653.  doi: 10.1081/PDE-120002868.  Google Scholar

[6]

P. Caro, P. Ola and M. Salo, Inverse boundary value problem for Maxwell equations with local data,, Comm. PDE, 34 (2009), 1425.  doi: 10.1080/03605300903296272.  Google Scholar

[7]

P. Caro, Stable determination of the electromagnetic coefficients by boundary measurements,, Inverse Problems, 26 (2010).   Google Scholar

[8]

D. Colton and L. Päivärinta, The uniqueness of a solution to an inverse scattering problem for electromagnetic waves,, Arch. Rational Mech. Anal., 119 (1992), 59.  doi: 10.1007/BF00376010.  Google Scholar

[9]

H. Heck and J.-N. Wang, Stability estimates for the inverse boundary value problem by partial Cauchy data,, Inverse Problems, 22 (2006), 1787.  doi: 10.1088/0266-5611/22/5/015.  Google Scholar

[10]

H. Heck and J.-N. Wang, Optimal stability estimate of the inverse boundary value problem by partial measurements,, preprint (2007) \arXiv{0708.3289v1}., (2007).   Google Scholar

[11]

V. Isakov, Carleman estimates and applications to inverse problems,, Milan J. Math., 72 (2004), 249.  doi: 10.1007/s00032-004-0033-6.  Google Scholar

[12]

V. Isakov, On uniqueness in the inverse conductivity problem with local data,, Inverse Probl. Imaging, 1 (2007), 95.   Google Scholar

[13]

D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains,, J. Funct. Anal., 130 (1995), 161.  doi: 10.1006/jfan.1995.1067.  Google Scholar

[14]

M. Joshi, S. R. McDowall, Total determination of material parameters from electromagnetic boundary information,, Pacific J. Math., 193 (2000), 107.  doi: 10.2140/pjm.2000.193.107.  Google Scholar

[15]

C. E. Kenig, M. Salo and G. Uhlmann, Inverse problems for the anisotropic Maxwell equations,, Duke Math. J., 157 (2011), 369.  doi: 10.1215/00127094-1272903.  Google Scholar

[16]

C. E. Kenig, J. Sjöstrand and G. Uhlmann, The Calderón problem with partial data,, Ann. of Math., 165 (2007), 567.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[17]

Y. Kurylev, M. Lassas, Matti and E. Somersalo, Maxwell's equations with a polarization independent wave velocity: Direct and inverse problems,, J. Math. Pures Appl., 86 (2006), 237.  doi: 10.1016/j.matpur.2006.01.008.  Google Scholar

[18]

R. Leis, "Initial Boundary Value Problems in Mathematical Physics,", Wiley, (1986).   Google Scholar

[19]

H. Liu, M. Yamamoto and J. Zou, Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering,, Inverse Problems, 23 (2007), 2357.  doi: 10.1088/0266-5611/23/6/005.  Google Scholar

[20]

S. R. McDowall, An electromagnetic inverse problem in chiral media,, Trans. Amer. Math. Soc., 352 (2000), 2993.  doi: 10.1090/S0002-9947-00-02518-6.  Google Scholar

[21]

M. Mitrea, Sharp Hodge decomposition, Maxwell's equations, and vector Poisson problems on non-smooth, three-dimensional riemannian manifolds,, Duke Math. J., 125 (2004), 467.  doi: 10.1215/S0012-7094-04-12322-1.  Google Scholar

[22]

P. Ola, L. Päivärinta and E. Somersalo, An inverse boundary value problem in electrodynamics,, Duke Math. J., 70 (1993), 617.  doi: 10.1215/S0012-7094-93-07014-7.  Google Scholar

[23]

P. Ola, L. Päivärinta and E. Somersalo, Inverse problems for time harmonic electrodynamics. Inside out: inverse problems and applications,, 169-191, 47 (2003), 169.   Google Scholar

[24]

P. Ola and E. Somersalo, Electromagnetic inverse problems and generalized Sommerfeld potentials,, SIAM J. Appl. Math., 56 (1996), 1129.  doi: 10.1137/S0036139995283948.  Google Scholar

[25]

M. Salo and L. Tzou, Carleman estimates and inverse problems for Dirac operators,, Math. Ann., 344 (2009), 161.  doi: 10.1007/s00208-008-0301-9.  Google Scholar

[26]

M. Salo and L. Tzou, Inverse problems with partial data for a Dirac system: A Carleman estimate approach,, Adv. Math., 225 (2010), 487.  doi: 10.1016/j.aim.2010.03.003.  Google Scholar

[27]

E. Somersalo, D. Isaacson and M. Cheney, A linearized inverse boundary value problem for Maxwell's equations,, J. Comp. Appl. Math., 42 (1992), 123.  doi: 10.1016/0377-0427(92)90167-V.  Google Scholar

[28]

E. Stein, "Singular Integrals and Differentiability Properties of Functions,", Princeton University Press, (1970).   Google Scholar

[29]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem,, Ann. of Math., 125 (1987), 153.  doi: 10.2307/1971291.  Google Scholar

[30]

H. Triebel, Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers,, Rev. Mat. Complut., 15 (2002), 475.   Google Scholar

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