doi: 10.3934/eect.2021060
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Stability estimate for a partial data inverse problem for the convection-diffusion equation

1. 

TIFR Centre for Applicable Mathematics, Bangalore 560065, India

2. 

Department of Mathematics, Indian Institute of Technology, Jammu 181221, India

*Corresponding author: Manmohan Vashisth

Received  April 2021 Revised  September 2021 Early access December 2021

In this article, we study the stability in the inverse problem of determining the time-dependent convection term and density coefficient appearing in the convection-diffusion equation, from partial boundary measurements. For dimension $ n\ge 2 $, we show the convection term (modulo the gauge term) admits log-log stability, whereas log-log-log stability estimate is obtained for the density coefficient.

Citation: Soumen Senapati, Manmohan Vashisth. Stability estimate for a partial data inverse problem for the convection-diffusion equation. Evolution Equations and Control Theory, doi: 10.3934/eect.2021060
References:
[1]

S. A. Avdonin and M. I. Belishev, Dynamical inverse problem for the Schrödinger equation (BC-method), Proceedings of the St. Petersburg Mathematical Society, Amer. Math. Soc. Transl. Ser. Vol. X, 214 (2005), 1-14.  doi: 10.1090/trans2/214/01.

[2]

S. A. Avdonin and T. I. Seidman, Identication of $q(x)$ in $u_t = \Delta u - qu$, from boundary observations, SIAM J. Control Optim., 33 (1995), 1247-1255.  doi: 10.1137/S0363012993249729.

[3]

L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 23 (2007), 1327-1328.  doi: 10.1088/0266-5611/23/3/C01.

[4]

M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.

[5]

M. Bellassoued and I. Ben Aïcha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, J. Math. Anal. Appl., 449 (2017), 46-76.  doi: 10.1016/j.jmaa.2016.11.082.

[6]

M. Bellassoued and O. Ben Fraj, Stably determining time-dependent convection-diffusion coefficients from a partial Dirichlet-to-Neumann map, Inverse Problems, 37 (2021), 045011, 35pp. doi: 10.1088/1361-6420/abe10d.

[7]

M. Bellassoued and D. Dos Santos Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010, 30pp. doi: 10.1088/0266-5611/26/12/125010.

[8]

M. BellassouedD. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243.  doi: 10.1080/00036810600787873.

[9]

M. BellassouedD. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046.  doi: 10.1016/j.jmaa.2008.01.098.

[10]

M. BellassouedY. Kian and E. Soccorsi, An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562.  doi: 10.1016/j.jde.2016.01.033.

[11]

M. BellassouedY. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci., 54 (2018), 679-728.  doi: 10.4171/PRIMS/54-4-1.

[12]

M. Bellassoued and I. Rassas, Stability estimate for an inverse problem of the convection-diffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 71-92.  doi: 10.1515/jiip-2018-0072.

[13]

I. Ben Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. doi: 10.1088/0266-5611/31/12/125010.

[14]

I. Ben Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, J. Math. Phys., 58 (2017), 071508, 21pp. doi: 10.1063/1.4995606.

[15]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large class of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269–272.

[16]

A. L. Bukhge${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$m and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.

[17]

P. Caro and Y. Kian, Determination of convection terms and quasi-linearities appearing in diffusion equations, preprint, arXiv: 1812.08495.

[18]

J. Cheng and M. Yamamoto, The global uniqueness for determining two convection coefficients from Dirichlet to Neumann map in two dimensions, Inverse Problems, 16 (2000), L25–L30. doi: 10.1088/0266-5611/16/3/101.

[19]

J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Anal., 50 (2002), 163-171.  doi: 10.1016/S0362-546X(01)00742-8.

[20]

J. Cheng and M. Yamamoto, Determination of Two Convection Coefficients from Dirichlet to Neumann Map in the Two-Dimensional Case, SIAM J. Math. Anal., 35 (2004), 1371-1393.  doi: 10.1137/S0036141003422497.

[21]

M. Choulli, An abstract inverse problem, J. Appl. Math. Stoc. Ana., 4 (1991), 117-128.  doi: 10.1155/S1048953391000084.

[22]

M. Choulli, An abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202.  doi: 10.1016/0022-247X(91)90299-F.

[23]

M. Choulli, Une Introduction Aux problèmes Inverses Elliptiques et Paraboliques, , Mathématiques & Applications, 65. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.

[24]

M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160.  doi: 10.3934/mcrf.2013.3.143.

[25]

M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261.  doi: 10.1016/j.matpur.2017.12.003.

[26]

M. ChoulliY. Kian and E. Soccorsi, Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.  doi: 10.1137/140986268.

[27]

Z.-C. DengJ.-N. Yu and Y. Liu, Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. Comput. Simulation, 77 (2008), 421-435.  doi: 10.1016/j.matcom.2008.01.002.

[28]

D. Dos Santos FerreiraC. E. KenigJ. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial {C}auchy data, Comm. Math. Phys., 271 (2007), 467-488.  doi: 10.1007/s00220-006-0151-9.

[29]

G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105, 18pp. doi: 10.1063/1.2841329.

[30]

P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. doi: 10.1088/0266-5611/29/6/065006.

[31]

G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging, 12 (2018), 745-772.  doi: 10.3934/ipi.2018032.

[32]

V. Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316.  doi: 10.1016/0022-0396(91)90051-A.

[33]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12.  doi: 10.1007/BF00392201.

[34]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127. Springer-Verlag, New York, 1998.

[35]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001.

[36]

Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.  doi: 10.3934/ipi.2014.8.713.

[37]

Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408–428. doi: 10.1016/j.jmaa.2015.12.018.

[38]

Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021–4046. doi: 10.1137/16M1076708.

[39]

Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 973–990. doi: 10.1016/j.anihpc.2016.07.003.

[40]

Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2019), 5087-5126.  doi: 10.1093/imrn/rnx263.

[41]

Y. Kian, Q. Sang Phan and E. Soccorsi, A Carleman estimate for infinite cyclindrical quantum domains and the application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp. doi: 10.1088/0266-5611/30/5/055016.

[42]

Y. Kian, Q. Sang Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, J. Math. Anal. Appl., 426 (2015), 194–210. doi: 10.1016/j.jmaa.2015.01.028.

[43]

Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.  doi: 10.1137/18M1197308.

[44]

Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839.  doi: 10.3934/ipi.2020038.

[45]

Y. Kian and M. Yamamoto, Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations, Inverse Problems, 35 (2019), 115006, 24pp doi: 10.1088/1361-6420/ab2d42.

[46]

V. P. Krishnan and M. Vashisth, An inverse problem for the relativistic Schrödinger equation with partial boundary data, Appl. Anal., 99 (2020), 1889-1909.  doi: 10.1080/00036811.2018.1549321.

[47]

J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. 3. (French) Travaux et Recherches Mathématiques, Dunod, Paris, 1970.

[48]

R. K. Mishra and M. Vashisth, Determining the Time Dependent Matrix Potential in A Wave Equation From Partial Boundary Data, Appl. Anal., 100 (2021), 3492-3508.  doi: 10.1080/00036811.2020.1721476.

[49]

N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42. 

[50]

G. Nakamura and S. Sasayama, Inverse boundary value problem for the heat equation with discontinuous coefficients, J. Inverse Ill-Posed Probl., 21 (2013), 217-232.  doi: 10.1515/jip-2012-0073.

[51]

V. Pohjola, A uniqueness result for an inverse problem of the steady state convection-diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103.  doi: 10.1137/140970926.

[52]

Ra kesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96.  doi: 10.1080/03605308808820539.

[53]

S. K. Sahoo and M. Vashisth, A partial data inverse problem for Convection-Diffusion equation, Inverse Probl. Imaging, 14 (2020), 53-75.  doi: 10.3934/ipi.2019063.

[54]

R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. doi: 10.1088/0266-5611/29/9/095015.

[55]

S. Senapati, Stability estimates for the relativistic Schrödinger equation from partial boundary data, Inverse Problems, 37 (2021), 015001, 25pp. doi: 10.1088/1361-6420/abca7f.

[56]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[57]

S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), 695-703.  doi: 10.1515/form.1999.020.

show all references

References:
[1]

S. A. Avdonin and M. I. Belishev, Dynamical inverse problem for the Schrödinger equation (BC-method), Proceedings of the St. Petersburg Mathematical Society, Amer. Math. Soc. Transl. Ser. Vol. X, 214 (2005), 1-14.  doi: 10.1090/trans2/214/01.

[2]

S. A. Avdonin and T. I. Seidman, Identication of $q(x)$ in $u_t = \Delta u - qu$, from boundary observations, SIAM J. Control Optim., 33 (1995), 1247-1255.  doi: 10.1137/S0363012993249729.

[3]

L. Baudouin and J.-P. Puel, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 23 (2007), 1327-1328.  doi: 10.1088/0266-5611/23/3/C01.

[4]

M. I. Belishev, Recent progress in the boundary control method, Inverse Problems, 23 (2007), R1–R67. doi: 10.1088/0266-5611/23/5/R01.

[5]

M. Bellassoued and I. Ben Aïcha, Stable determination outside a cloaking region of two time-dependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, J. Math. Anal. Appl., 449 (2017), 46-76.  doi: 10.1016/j.jmaa.2016.11.082.

[6]

M. Bellassoued and O. Ben Fraj, Stably determining time-dependent convection-diffusion coefficients from a partial Dirichlet-to-Neumann map, Inverse Problems, 37 (2021), 045011, 35pp. doi: 10.1088/1361-6420/abe10d.

[7]

M. Bellassoued and D. Dos Santos Ferreira, Stable determination of coefficients in the dynamical anisotropic Schrödinger equation from the Dirichlet-to-Neumann map, Inverse Problems, 26 (2010), 125010, 30pp. doi: 10.1088/0266-5611/26/12/125010.

[8]

M. BellassouedD. Jellali and M. Yamamoto, Lipschitz stability for a hyperbolic inverse problem by finite local boundary data, Appl. Anal., 85 (2006), 1219-1243.  doi: 10.1080/00036810600787873.

[9]

M. BellassouedD. Jellali and M. Yamamoto, Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map, J. Math. Anal. Appl., 343 (2008), 1036-1046.  doi: 10.1016/j.jmaa.2008.01.098.

[10]

M. BellassouedY. Kian and E. Soccorsi, An inverse stability result for non-compactly supported potentials by one arbitrary lateral Neumann observation, J. Differential Equations, 260 (2016), 7535-7562.  doi: 10.1016/j.jde.2016.01.033.

[11]

M. BellassouedY. Kian and E. Soccorsi, An inverse problem for the magnetic Schrödinger equation in infinite cylindrical domains, Publ. Res. Inst. Math. Sci., 54 (2018), 679-728.  doi: 10.4171/PRIMS/54-4-1.

[12]

M. Bellassoued and I. Rassas, Stability estimate for an inverse problem of the convection-diffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 71-92.  doi: 10.1515/jiip-2018-0072.

[13]

I. Ben Aïcha, Stability estimate for a hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010, 21pp. doi: 10.1088/0266-5611/31/12/125010.

[14]

I. Ben Aïcha, Stability estimate for an inverse problem for the Schrödinger equation in a magnetic field with time-dependent coefficient, J. Math. Phys., 58 (2017), 071508, 21pp. doi: 10.1063/1.4995606.

[15]

A. L. Bukhgeĭm and M. V. Klibanov, Uniqueness in the large class of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), 269–272.

[16]

A. L. Bukhge${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}}$m and G. Uhlmann, Recovering a potential from partial Cauchy data, Comm. Partial Differential Equations, 27 (2002), 653-668.  doi: 10.1081/PDE-120002868.

[17]

P. Caro and Y. Kian, Determination of convection terms and quasi-linearities appearing in diffusion equations, preprint, arXiv: 1812.08495.

[18]

J. Cheng and M. Yamamoto, The global uniqueness for determining two convection coefficients from Dirichlet to Neumann map in two dimensions, Inverse Problems, 16 (2000), L25–L30. doi: 10.1088/0266-5611/16/3/101.

[19]

J. Cheng and M. Yamamoto, Identification of convection term in a parabolic equation with a single measurement, Nonlinear Anal., 50 (2002), 163-171.  doi: 10.1016/S0362-546X(01)00742-8.

[20]

J. Cheng and M. Yamamoto, Determination of Two Convection Coefficients from Dirichlet to Neumann Map in the Two-Dimensional Case, SIAM J. Math. Anal., 35 (2004), 1371-1393.  doi: 10.1137/S0036141003422497.

[21]

M. Choulli, An abstract inverse problem, J. Appl. Math. Stoc. Ana., 4 (1991), 117-128.  doi: 10.1155/S1048953391000084.

[22]

M. Choulli, An abstract inverse problem and application, J. Math. Anal. Appl., 160 (1991), 190-202.  doi: 10.1016/0022-247X(91)90299-F.

[23]

M. Choulli, Une Introduction Aux problèmes Inverses Elliptiques et Paraboliques, , Mathématiques & Applications, 65. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.

[24]

M. Choulli and Y. Kian, Stability of the determination of a time-dependent coefficient in parabolic equations, Math. Control Relat. Fields, 3 (2013), 143-160.  doi: 10.3934/mcrf.2013.3.143.

[25]

M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261.  doi: 10.1016/j.matpur.2017.12.003.

[26]

M. ChoulliY. Kian and E. Soccorsi, Stable determination of time-dependent scalar potential from boundary measurements in a periodic quantum waveguide, SIAM J. Math. Anal., 47 (2015), 4536-4558.  doi: 10.1137/140986268.

[27]

Z.-C. DengJ.-N. Yu and Y. Liu, Identifying the coefficient of first-order in parabolic equation from final measurement data, Math. Comput. Simulation, 77 (2008), 421-435.  doi: 10.1016/j.matcom.2008.01.002.

[28]

D. Dos Santos FerreiraC. E. KenigJ. Sjöstrand and G. Uhlmann, Determining a magnetic Schrödinger operator from partial {C}auchy data, Comm. Math. Phys., 271 (2007), 467-488.  doi: 10.1007/s00220-006-0151-9.

[29]

G. Eskin, Inverse problems for the Schrödinger equations with time-dependent electromagnetic potentials and the Aharonov-Bohm effect, J. Math. Phys., 49 (2008), 022105, 18pp. doi: 10.1063/1.2841329.

[30]

P. Gaitan and Y. Kian, A stability result for a time-dependent potential in a cylindrical domain, Inverse Problems, 29 (2013), 065006, 18pp. doi: 10.1088/0266-5611/29/6/065006.

[31]

G. Hu and Y. Kian, Determination of singular time-dependent coefficients for wave equations from full and partial data, Inverse Probl. Imaging, 12 (2018), 745-772.  doi: 10.3934/ipi.2018032.

[32]

V. Isakov, Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316.  doi: 10.1016/0022-0396(91)90051-A.

[33]

V. Isakov, On uniqueness in inverse problems for semilinear parabolic equations, Arch. Rational Mech. Anal., 124 (1993), 1-12.  doi: 10.1007/BF00392201.

[34]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127. Springer-Verlag, New York, 1998.

[35]

A. Katchalov, Y. Kurylev and M. Lassas, Inverse Boundary Spectral Problems, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 123, Chapman & Hall/CRC, Boca Raton, FL, 2001.

[36]

Y. Kian, Stability of the determination of a coefficient for wave equations in an infinite waveguide, Inverse Probl. Imaging, 8 (2014), 713-732.  doi: 10.3934/ipi.2014.8.713.

[37]

Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408–428. doi: 10.1016/j.jmaa.2015.12.018.

[38]

Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equations from partial data, SIAM J. Math. Anal., 48 (2016), 4021–4046. doi: 10.1137/16M1076708.

[39]

Y. Kian, Unique determination of a time-dependent potential for wave equations from partial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 973–990. doi: 10.1016/j.anihpc.2016.07.003.

[40]

Y. Kian and L. Oksanen, Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations, Int. Math. Res. Not. IMRN, 2019 (2019), 5087-5126.  doi: 10.1093/imrn/rnx263.

[41]

Y. Kian, Q. Sang Phan and E. Soccorsi, A Carleman estimate for infinite cyclindrical quantum domains and the application to inverse problems, Inverse Problems, 30 (2014), 055016, 16 pp. doi: 10.1088/0266-5611/30/5/055016.

[42]

Y. Kian, Q. Sang Phan and E. Soccorsi, Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains, J. Math. Anal. Appl., 426 (2015), 194–210. doi: 10.1016/j.jmaa.2015.01.028.

[43]

Y. Kian and E. Soccorsi, Hölder stably determining the time-dependent electromagnetic potential of the Schrödinger equation, SIAM J. Math. Anal., 51 (2019), 627-647.  doi: 10.1137/18M1197308.

[44]

Y. Kian and A. Tetlow, Hölder stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic schrödinger equation, Inverse Probl. Imaging, 14 (2020), 819-839.  doi: 10.3934/ipi.2020038.

[45]

Y. Kian and M. Yamamoto, Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations, Inverse Problems, 35 (2019), 115006, 24pp doi: 10.1088/1361-6420/ab2d42.

[46]

V. P. Krishnan and M. Vashisth, An inverse problem for the relativistic Schrödinger equation with partial boundary data, Appl. Anal., 99 (2020), 1889-1909.  doi: 10.1080/00036811.2018.1549321.

[47]

J.-L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, Vol. 3. (French) Travaux et Recherches Mathématiques, Dunod, Paris, 1970.

[48]

R. K. Mishra and M. Vashisth, Determining the Time Dependent Matrix Potential in A Wave Equation From Partial Boundary Data, Appl. Anal., 100 (2021), 3492-3508.  doi: 10.1080/00036811.2020.1721476.

[49]

N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, (Russian) Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42. 

[50]

G. Nakamura and S. Sasayama, Inverse boundary value problem for the heat equation with discontinuous coefficients, J. Inverse Ill-Posed Probl., 21 (2013), 217-232.  doi: 10.1515/jip-2012-0073.

[51]

V. Pohjola, A uniqueness result for an inverse problem of the steady state convection-diffusion equation, SIAM J. Math. Anal., 47 (2015), 2084-2103.  doi: 10.1137/140970926.

[52]

Ra kesh and W. W. Symes, Uniqueness for an inverse problem for the wave equation, Comm. Partial Differential Equations, 13 (1988), 87-96.  doi: 10.1080/03605308808820539.

[53]

S. K. Sahoo and M. Vashisth, A partial data inverse problem for Convection-Diffusion equation, Inverse Probl. Imaging, 14 (2020), 53-75.  doi: 10.3934/ipi.2019063.

[54]

R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp. doi: 10.1088/0266-5611/29/9/095015.

[55]

S. Senapati, Stability estimates for the relativistic Schrödinger equation from partial boundary data, Inverse Problems, 37 (2021), 015001, 25pp. doi: 10.1088/1361-6420/abca7f.

[56]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

[57]

S. Vessella, A continuous dependence result in the analytic continuation problem, Forum Math., 11 (1999), 695-703.  doi: 10.1515/form.1999.020.

[1]

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