February  2022, 16(1): 251-281. doi: 10.3934/ipi.2021049

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

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 Published  February 2022 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.

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 and Imaging, 2022, 16 (1) : 251-281. doi: 10.3934/ipi.2021049
References:
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G. Alessandrini, Stable determination of conductivity by boundary measurements, Applicable Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[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.

[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.

[4]

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

[5]

G. BaoJ. 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.

[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.

[7]

E. BerettaM. V. De HoopF. 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.

[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.

[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.

[10]

J. ChengV. 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.

[11]

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

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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.

[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.

[14]

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

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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.

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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.

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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.

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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.

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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.

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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.

[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.

[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.

[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.

[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.

[25]

V. IsakovR.-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.

[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.

[27]

V. IsakovS. NagayasuG. 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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[37]

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

[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.

[39]

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

[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.

[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. 

[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.

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

[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.

[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.

[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.

[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.

[49]

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

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[5]

G. BaoJ. 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.

[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.

[7]

E. BerettaM. V. De HoopF. 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.

[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.

[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.

[10]

J. ChengV. 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.

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

V. IsakovR.-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.

[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.

[27]

V. IsakovS. NagayasuG. 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.

[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.

[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.

[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.

[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.

[32]

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

[33]

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

[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.

[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.

[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.

[37]

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

[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.

[39]

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

[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.

[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. 

[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.

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

[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.

[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.

[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.

[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.

[49]

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

[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.

[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.

[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.

[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.

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Sombuddha Bhattacharyya. An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data. Inverse Problems and Imaging, 2018, 12 (3) : 801-830. doi: 10.3934/ipi.2018034

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