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doi: 10.3934/ipi.2021049
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Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

María Ángeles García-Ferrero 1,, , Angkana Rüland 1, and  Wiktoria Zatoń 2,

1. 

Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
2. 

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

* Corresponding author: garciaferrero@uni-heidelberg.de

Received  January 2021 Revised  May 2021 Early access July 2021

In this article, we discuss quantitative Runge approximation properties for the acoustic Helmholtz equation and prove stability improvement results in the high frequency limit for an associated partial data inverse problem modelled on [3,35]. The results rely on quantitative unique continuation estimates in suitable function spaces with explicit frequency dependence. We contrast the frequency dependence of interior Runge approximation results from non-convex and convex sets.

Keywords: Runge approximation, stability estimates, partial data inverse problem, high frequency limit, quantitative unique continuation.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35J05.
Citation: María Ángeles García-Ferrero, Angkana Rüland, Wiktoria Zatoń. Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation. Inverse Problems & Imaging, doi: 10.3934/ipi.2021049
References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[2]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004.  Google Scholar

[3]

H. Ammari and G. Uhlmann, Reconstruction of the potential from partial Cauchy data for the Schrödinger equation, Indiana University Mathematics Journal, 53 (2004), 169-183.  doi: 10.1512/iumj.2004.53.2299.  Google Scholar

[4]

L. Bakri, Quantitative uniqueness for Schrödinger operator, Indiana University Mathematics Journal, 61 (2012), 1565-1580.  doi: 10.1512/iumj.2012.61.4713.  Google Scholar

[5]

G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, Journal of Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar

[6]

G. Bao, P. Li and Y. Zhao, Stability for the inverse source problems in elastic and electromagnetic waves, Journal de Mathématiques Pures et Appliquées, 134 (2020), 122–178. doi: 10.1016/j.matpur.2019.06.006.  Google Scholar

[7]

E. Beretta, M. V. De Hoop, F. Faucher and O. Scherzer, Inverse boundary value problem for the Helmholtz equation: Quantitative conditional Lipschitz stability estimates, SIAM Journal on Mathematical Analysis, 48 (2016), 3962-3983.  doi: 10.1137/15M1043856.  Google Scholar

[8]

S. M. Berge and E. Malinnikova, On the three ball theorem for solutions of the Helmholtz equation, Complex Analysis and its Synergies, 7 (2021), 14. doi: 10.1007/s40627-021-00070-3.  Google Scholar

[9]

E. Burman, M. Nechita and L. Oksanen, Unique continuation for the Helmholtz equation using stabilized finite element methods, Journal de Mathématiques Pures et Appliquées, 129 (2019), 1–22. doi: 10.1016/j.matpur.2018.10.003.  Google Scholar

[10]

J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, Journal of Differential Equations, 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030.  Google Scholar

[11]

A. Enciso and D. Peralta-Salas, Approximation theorems for the Schrödinger equation and quantum vortex reconnection, arXiv: 1905.02467. Google Scholar

[12]

M. N. Entekhabi and V. Isakov, On increasing stability in the two dimensional inverse source scattering problem with many frequencies, Inverse Problems, 34 (2018), 055005. doi: 10.1088/1361-6420/aab465.  Google Scholar

[13]

M. Entekhabi and V. Isakov, Increasing stability in acoustic and elastic inverse source problems, SIAM Journal on Mathematical Analysis, 52 (2020), 5232-5256.  doi: 10.1137/19M1279885.  Google Scholar

[14]

L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Springer-Verlag, Berlin, 1994.  Google Scholar

[15]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Hemholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004.  Google Scholar

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M. I. Isaev, Exponential instability in the inverse scattering problem on the energy interval, Functional Analysis and its Applications, 47 (2013), 187-194.  doi: 10.1007/s10688-013-0025-9.  Google Scholar

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M. I. Isaev, Instability in the Gel'fand inverse problem at high energies, Applicable Analysis, 92 (2013), 2262-2274.  doi: 10.1080/00036811.2012.731501.  Google Scholar

[18]

M. I. Isaev and R. G. Novikov, Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions, Journal of Inverse and Ill-Posed Problems, 20 (2012), 313-325.  doi: 10.1515/jip-2012-0024.  Google Scholar

[19]

M. I. Isaev and R. G. Novikov, Effectivized Hölder-logarithmic stability estimates for the Gel'fand inverse problem, Inverse Problems, 30 (2014), 095006. doi: 10.1088/0266-5611/30/9/095006.  Google Scholar

[20]

M. I. Isaev and R. G. Novikov, Stability estimates for recovering the potential by the impedance boundary map, St. Petersburg Mathematical Journal, 25 (2014), 23-41.  doi: 10.1090/s1061-0022-2013-01278-7.  Google Scholar

[21]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemporary Mathematics, 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.  Google Scholar

[22]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discrete & Continuous Dynamical Systems-S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.  Google Scholar

[23]

V. Isakov, On increasing stability of the continuation for elliptic equations of second order without (pseudo) convexity assumptions, Inverse Problems & Imaging, 13 (2019), 983-1006.  doi: 10.3934/ipi.2019044.  Google Scholar

[24]

V. Isakov and S. Kindermann, Subspaces of stability in the Cauchy problem for the Helmholtz equation, Methods and Applications of Analysis, 18 (2011), 1-30.  doi: 10.4310/MAA.2011.v18.n1.a1.  Google Scholar

[25]

V. Isakov, R.-Y. Lai and J.-N. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM Journal on Mathematical Analysis, 48 (2016), 569-594.  doi: 10.1137/15M1019052.  Google Scholar

[26]

V. Isakov and S. Lu, Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM Journal on Applied Mathematics, 78 (2018), 1-18.  doi: 10.1137/17M1112704.  Google Scholar

[27]

V. Isakov, S. Nagayasu, G. Uhlmann and J.-N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemporary Mathematics, 615 (2014), 131-141.  doi: 10.1090/conm/615/12268.  Google Scholar

[28]

V. Isakov and J.-N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Problems & Imaging, 8 (2014), 1139-1150.  doi: 10.3934/ipi.2014.8.1139.  Google Scholar

[29]

V. Isakov and J.-N. Wang, Uniqueness and increasing stability in electromagnetic inverse source problems, Journal of Differential Equations, 283 (2021), 110-135.  doi: 10.1016/j.jde.2021.02.035.  Google Scholar

[30]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, Journal of Functional Analysis, 130 (1995), 161-219.  doi: 10.1006/jfan.1995.1067.  Google Scholar

[31]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Communications on Pure and Applied Mathematics, 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402.  Google Scholar

[32]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin, 1995.  Google Scholar

[33]

H. Koch, A. Rüland and M. Salo, On instability mechanisms for inverse problems, arXiv: 2012.01855. Google Scholar

[34]

H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues, Communications in Mathematical Physics, 267 (2006), 419-449.  doi: 10.1007/s00220-006-0060-y.  Google Scholar

[35]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, Journal de Mathématiques Pures et Appliquées, 126 (2019), 273–291. doi: 10.1016/j.matpur.2019.02.017.  Google Scholar

[36]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[37]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[38]

A. Logunov, Nodal sets of Laplace eigenfunctions: Polynomial upper estimates of the Hausdorff measure, Annals of Mathematics, 187 (2018), 221-239.  doi: 10.4007/annals.2018.187.1.4.  Google Scholar

[39]

A. Logunov, E. Malinnikova, N. Nadirashvili and F. Nazarov, The Landis conjecture on exponential decay, arXiv: 2007.07034. Google Scholar

[40]

V. Z. Meshkov, On the possible rate of decay at infinity of solutions of second order partial differential equations, Matematicheskii Sbornik, 182 (1991), 364-383.  doi: 10.1070/SM1992v072n02ABEH001414.  Google Scholar

[41]

N. G. Meyers, An $ L^{p} $-estimate for the gradient of solutions of second order elliptic divergence equations, Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 17 (1963), 189-206.   Google Scholar

[42]

S. Nagayasu, G. Uhlmann and J.-N. Wang, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems, 29 (2013), 025012. doi: 10.1088/0266-5611/29/2/025012.  Google Scholar

[43] F. W. J. Olver, NIST Handbook of Mathematical Functions Hardback and CD-ROM, Cambridge University Press, 2010.   Google Scholar
[44]

R. B. Paris, An inequality for the Bessel function ${J}_\nu(\nu x)$, SIAM Journal on Mathematical Analysis, 15 (1984), 203-205.  doi: 10.1137/0515016.  Google Scholar

[45]

A. Rüland, Unique continuation for sublinear elliptic equations based on Carleman estimates, Journal of Differential Equations, 265 (2018), 6009-6035.  doi: 10.1016/j.jde.2018.07.025.  Google Scholar

[46]

A. Rüland and M. Salo, Quantitative Runge approximation and inverse problems, International Mathematics Research Notice, 2019 (2019), 6216-6234.  doi: 10.1093/imrn/rnx301.  Google Scholar

[47]

A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Analysis, 193 (2020), 111529. doi: 10.1016/j.na.2019.05.010.  Google Scholar

[48]

A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Mathematical Control & Related Fields, 10 (2020), 1-26.  doi: 10.3934/mcrf.2019027.  Google Scholar

[49]

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag Berlin, 1987. doi: 10.1007/978-3-642-96854-9.  Google Scholar

[50]

D. A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1697.  doi: 10.1088/0266-5611/23/4/019.  Google Scholar

[51]

D. A. Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), 074005. doi: 10.1088/0266-5611/26/7/074005.  Google Scholar

[52]

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

[53]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

show all references

References:
[1]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.  Google Scholar

[2]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004.  Google Scholar

[3]

H. Ammari and G. Uhlmann, Reconstruction of the potential from partial Cauchy data for the Schrödinger equation, Indiana University Mathematics Journal, 53 (2004), 169-183.  doi: 10.1512/iumj.2004.53.2299.  Google Scholar

[4]

L. Bakri, Quantitative uniqueness for Schrödinger operator, Indiana University Mathematics Journal, 61 (2012), 1565-1580.  doi: 10.1512/iumj.2012.61.4713.  Google Scholar

[5]

G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, Journal of Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar

[6]

G. Bao, P. Li and Y. Zhao, Stability for the inverse source problems in elastic and electromagnetic waves, Journal de Mathématiques Pures et Appliquées, 134 (2020), 122–178. doi: 10.1016/j.matpur.2019.06.006.  Google Scholar

[7]

E. Beretta, M. V. De Hoop, F. Faucher and O. Scherzer, Inverse boundary value problem for the Helmholtz equation: Quantitative conditional Lipschitz stability estimates, SIAM Journal on Mathematical Analysis, 48 (2016), 3962-3983.  doi: 10.1137/15M1043856.  Google Scholar

[8]

S. M. Berge and E. Malinnikova, On the three ball theorem for solutions of the Helmholtz equation, Complex Analysis and its Synergies, 7 (2021), 14. doi: 10.1007/s40627-021-00070-3.  Google Scholar

[9]

E. Burman, M. Nechita and L. Oksanen, Unique continuation for the Helmholtz equation using stabilized finite element methods, Journal de Mathématiques Pures et Appliquées, 129 (2019), 1–22. doi: 10.1016/j.matpur.2018.10.003.  Google Scholar

[10]

J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problem with many frequencies, Journal of Differential Equations, 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030.  Google Scholar

[11]

A. Enciso and D. Peralta-Salas, Approximation theorems for the Schrödinger equation and quantum vortex reconnection, arXiv: 1905.02467. Google Scholar

[12]

M. N. Entekhabi and V. Isakov, On increasing stability in the two dimensional inverse source scattering problem with many frequencies, Inverse Problems, 34 (2018), 055005. doi: 10.1088/1361-6420/aab465.  Google Scholar

[13]

M. Entekhabi and V. Isakov, Increasing stability in acoustic and elastic inverse source problems, SIAM Journal on Mathematical Analysis, 52 (2020), 5232-5256.  doi: 10.1137/19M1279885.  Google Scholar

[14]

L. Hörmander, The Analysis of Linear Partial Differential Operators IV, Springer-Verlag, Berlin, 1994.  Google Scholar

[15]

T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Hemholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004.  Google Scholar

[16]

M. I. Isaev, Exponential instability in the inverse scattering problem on the energy interval, Functional Analysis and its Applications, 47 (2013), 187-194.  doi: 10.1007/s10688-013-0025-9.  Google Scholar

[17]

M. I. Isaev, Instability in the Gel'fand inverse problem at high energies, Applicable Analysis, 92 (2013), 2262-2274.  doi: 10.1080/00036811.2012.731501.  Google Scholar

[18]

M. I. Isaev and R. G. Novikov, Energy and regularity dependent stability estimates for the Gel'fand inverse problem in multidimensions, Journal of Inverse and Ill-Posed Problems, 20 (2012), 313-325.  doi: 10.1515/jip-2012-0024.  Google Scholar

[19]

M. I. Isaev and R. G. Novikov, Effectivized Hölder-logarithmic stability estimates for the Gel'fand inverse problem, Inverse Problems, 30 (2014), 095006. doi: 10.1088/0266-5611/30/9/095006.  Google Scholar

[20]

M. I. Isaev and R. G. Novikov, Stability estimates for recovering the potential by the impedance boundary map, St. Petersburg Mathematical Journal, 25 (2014), 23-41.  doi: 10.1090/s1061-0022-2013-01278-7.  Google Scholar

[21]

V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemporary Mathematics, 426 (2007), 255-267.  doi: 10.1090/conm/426/08192.  Google Scholar

[22]

V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discrete & Continuous Dynamical Systems-S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631.  Google Scholar

[23]

V. Isakov, On increasing stability of the continuation for elliptic equations of second order without (pseudo) convexity assumptions, Inverse Problems & Imaging, 13 (2019), 983-1006.  doi: 10.3934/ipi.2019044.  Google Scholar

[24]

V. Isakov and S. Kindermann, Subspaces of stability in the Cauchy problem for the Helmholtz equation, Methods and Applications of Analysis, 18 (2011), 1-30.  doi: 10.4310/MAA.2011.v18.n1.a1.  Google Scholar

[25]

V. Isakov, R.-Y. Lai and J.-N. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM Journal on Mathematical Analysis, 48 (2016), 569-594.  doi: 10.1137/15M1019052.  Google Scholar

[26]

V. Isakov and S. Lu, Increasing stability in the inverse source problem with attenuation and many frequencies, SIAM Journal on Applied Mathematics, 78 (2018), 1-18.  doi: 10.1137/17M1112704.  Google Scholar

[27]

V. Isakov, S. Nagayasu, G. Uhlmann and J.-N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemporary Mathematics, 615 (2014), 131-141.  doi: 10.1090/conm/615/12268.  Google Scholar

[28]

V. Isakov and J.-N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Problems & Imaging, 8 (2014), 1139-1150.  doi: 10.3934/ipi.2014.8.1139.  Google Scholar

[29]

V. Isakov and J.-N. Wang, Uniqueness and increasing stability in electromagnetic inverse source problems, Journal of Differential Equations, 283 (2021), 110-135.  doi: 10.1016/j.jde.2021.02.035.  Google Scholar

[30]

D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, Journal of Functional Analysis, 130 (1995), 161-219.  doi: 10.1006/jfan.1995.1067.  Google Scholar

[31]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Communications on Pure and Applied Mathematics, 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402.  Google Scholar

[32]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag Berlin, 1995.  Google Scholar

[33]

H. Koch, A. Rüland and M. Salo, On instability mechanisms for inverse problems, arXiv: 2012.01855. Google Scholar

[34]

H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues, Communications in Mathematical Physics, 267 (2006), 419-449.  doi: 10.1007/s00220-006-0060-y.  Google Scholar

[35]

K. Krupchyk and G. Uhlmann, Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, Journal de Mathématiques Pures et Appliquées, 126 (2019), 273–291. doi: 10.1016/j.matpur.2019.02.017.  Google Scholar

[36]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations, 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.  Google Scholar

[37]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.  doi: 10.1137/1030001.  Google Scholar

[38]

A. Logunov, Nodal sets of Laplace eigenfunctions: Polynomial upper estimates of the Hausdorff measure, Annals of Mathematics, 187 (2018), 221-239.  doi: 10.4007/annals.2018.187.1.4.  Google Scholar

[39]

A. Logunov, E. Malinnikova, N. Nadirashvili and F. Nazarov, The Landis conjecture on exponential decay, arXiv: 2007.07034. Google Scholar

[40]

V. Z. Meshkov, On the possible rate of decay at infinity of solutions of second order partial differential equations, Matematicheskii Sbornik, 182 (1991), 364-383.  doi: 10.1070/SM1992v072n02ABEH001414.  Google Scholar

[41]

N. G. Meyers, An $ L^{p} $-estimate for the gradient of solutions of second order elliptic divergence equations, Annali Della Scuola Normale Superiore di Pisa-Classe di Scienze, 17 (1963), 189-206.   Google Scholar

[42]

S. Nagayasu, G. Uhlmann and J.-N. Wang, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems, 29 (2013), 025012. doi: 10.1088/0266-5611/29/2/025012.  Google Scholar

[43] F. W. J. Olver, NIST Handbook of Mathematical Functions Hardback and CD-ROM, Cambridge University Press, 2010.   Google Scholar
[44]

R. B. Paris, An inequality for the Bessel function ${J}_\nu(\nu x)$, SIAM Journal on Mathematical Analysis, 15 (1984), 203-205.  doi: 10.1137/0515016.  Google Scholar

[45]

A. Rüland, Unique continuation for sublinear elliptic equations based on Carleman estimates, Journal of Differential Equations, 265 (2018), 6009-6035.  doi: 10.1016/j.jde.2018.07.025.  Google Scholar

[46]

A. Rüland and M. Salo, Quantitative Runge approximation and inverse problems, International Mathematics Research Notice, 2019 (2019), 6216-6234.  doi: 10.1093/imrn/rnx301.  Google Scholar

[47]

A. Rüland and M. Salo, The fractional Calderón problem: Low regularity and stability, Nonlinear Analysis, 193 (2020), 111529. doi: 10.1016/j.na.2019.05.010.  Google Scholar

[48]

A. Rüland and M. Salo, Quantitative approximation properties for the fractional heat equation, Mathematical Control & Related Fields, 10 (2020), 1-26.  doi: 10.3934/mcrf.2019027.  Google Scholar

[49]

M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag Berlin, 1987. doi: 10.1007/978-3-642-96854-9.  Google Scholar

[50]

D. A. Subbarayappa and V. Isakov, On increased stability in the continuation of the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1697.  doi: 10.1088/0266-5611/23/4/019.  Google Scholar

[51]

D. A. Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), 074005. doi: 10.1088/0266-5611/26/7/074005.  Google Scholar

[52]

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

[53]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.  doi: 10.1016/S1874-5717(07)80010-7.  Google Scholar

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