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October  2019, 13(5): 983-1006. doi: 10.3934/ipi.2019044

## On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions

 Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033, USA

Received  September 2018 Revised  May 2019 Published  July 2019

Fund Project: The author is supported by NSF grant 15-14886 and the Emylou Keith and Betty Dutcher Distinguished Professorship.

We derive bounds of solutions of the Cauchy problem for general elliptic partial differential equations of second order containing parameter (wave number) $k$ which are getting nearly Lipschitz for large $k$. Proofs use energy estimates combined with splitting solutions into low and high frequencies parts, an associated hyperbolic equation and the Fourier-Bros-Iagolnitzer transform to replace the hyperbolic equation with an elliptic equation without parameter $k$. The results suggest a better resolution in prospecting by various (acoustic, electromagnetic, etc) stationary waves with higher wave numbers without any geometric assumptions on domains and observation sites.

Citation: Victor Isakov. On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions. Inverse Problems and Imaging, 2019, 13 (5) : 983-1006. doi: 10.3934/ipi.2019044
##### References:
 [1] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability in the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004. [2] D. Aralumallige Subbarayappa and V. Isakov, On increased stability in the continuation for the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1697.  doi: 10.1088/0266-5611/23/4/019. [3] D. Aralumallige Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), 074005, 14pp. doi: 10.1088/0266-5611/26/7/074005. [4] R. Bosi, Y. Kurylev and M. Lassas, Stability of the unique continuation for the wave operator via Tataru's inequality and applications, J. Diff. Equat., 260 (2016), 6451-6492.  doi: 10.1016/j.jde.2015.12.043. [5] J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problems with many frequencies, J. Diff. Equat., 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030. [6] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and Stability in the Cauchy Problem for Maxwell' and elasticity systems, in Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vol. XIV (Paris, 1997/1998), North-Holland, Elsevier Science, 31 (2002), 329–349. doi: 10.1016/S0168-2024(02)80016-9. [7] T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004. [8] V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., AMS, 426 (2007), 255-267.  doi: 10.1090/conm/426/08192. [9] V. Isakov, Increased stability in the Cauchy problem for some elliptic equations, in Instability in Models Connected with Fluid Flow (eds. C. Bardos, A. Fursikov), Intern. Math. Series, Springer-Verlag, 6 (2008), 339–362. doi: 10.1007/978-0-387-75217-4_8. [10] V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discr. Cont. Dyn. Syst. S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631. [11] V. Isakov, Increasing stability of the continuation for general elliptic equations of second order, in New Prospects in Direct, Inverse, and Control Problems for Evolution Equations, Springer INdAM Series 10 (2014), 203–218. doi: 10.1007/978-3-319-11406-4_10. [12] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-51658-5. [13] V. Isakov and S. Kindermann, Regions of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. of Anal., 18 (2011), 1-29.  doi: 10.4310/MAA.2011.v18.n1.a1. [14] V. Isakov and S. Lu, Inverse source problems without (pseudo)convexity assumptions, Inv. Probl. Imag., 12 (2018), 955-970.  doi: 10.3934/ipi.2018040. [15] V. Isakov and S. Lu, Increasing stability in the inverse source problems with attenuation and many frequencies, SIAM J. Appl. Math., 78 (2018), 1-18.  doi: 10.1137/17M1112704. [16] V. Isakov, R.-Y. Lai and J.-N. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594.  doi: 10.1137/15M1019052. [17] V. Isakov and J.-N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to Neumann map, Inv. Probl. Imag., 8 (2014), 1139-1150.  doi: 10.3934/ipi.2014.8.1139. [18] F. John, Continuous Dependence on Data for Solutions of Partial Differential Equations With a Prescribed Bound, Comm. Pure Appl. Math., 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402. [19] L. Robbiano, Theoreme d'unicite adapte au controle des solutions des problemes hyperboliques, Comm. Part. Diff. Equat., 16 (1991), 789-800.  doi: 10.1080/03605309108820778. [20] L. Robbiano, Fonction de cout et controle des solutions des equations hyperboliques, Asympt. Anal., 10 (1995), 95-115. [21] L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-529.  doi: 10.1007/s002220050212. [22] D. Tataru, Unique continuation for solutions to PDE: Between Hörmander's Theorem and Holmgren's Theorem, Comm. Part. Diff. Equat., 20 (1995), 855-884.  doi: 10.1080/03605309508821117. [23] S. Vessella, Stability estimates for an inverse hyperbolic initial boundary value problem with unknown boundaries, SIAM J. Math. Anal., 47 (2016), 1419-1457.  doi: 10.1137/140976212. [24] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.

show all references

##### References:
 [1] G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability in the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004. [2] D. Aralumallige Subbarayappa and V. Isakov, On increased stability in the continuation for the Helmholtz equation, Inverse Problems, 23 (2007), 1689-1697.  doi: 10.1088/0266-5611/23/4/019. [3] D. Aralumallige Subbarayappa and V. Isakov, Increasing stability of the continuation for the Maxwell system, Inverse Problems, 26 (2010), 074005, 14pp. doi: 10.1088/0266-5611/26/7/074005. [4] R. Bosi, Y. Kurylev and M. Lassas, Stability of the unique continuation for the wave operator via Tataru's inequality and applications, J. Diff. Equat., 260 (2016), 6451-6492.  doi: 10.1016/j.jde.2015.12.043. [5] J. Cheng, V. Isakov and S. Lu, Increasing stability in the inverse source problems with many frequencies, J. Diff. Equat., 260 (2016), 4786-4804.  doi: 10.1016/j.jde.2015.11.030. [6] M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and Stability in the Cauchy Problem for Maxwell' and elasticity systems, in Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vol. XIV (Paris, 1997/1998), North-Holland, Elsevier Science, 31 (2002), 329–349. doi: 10.1016/S0168-2024(02)80016-9. [7] T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.  doi: 10.1088/0266-5611/20/3/004. [8] V. Isakov, Increased stability in the continuation for the Helmholtz equation with variable coefficient, Contemp. Math., AMS, 426 (2007), 255-267.  doi: 10.1090/conm/426/08192. [9] V. Isakov, Increased stability in the Cauchy problem for some elliptic equations, in Instability in Models Connected with Fluid Flow (eds. C. Bardos, A. Fursikov), Intern. Math. Series, Springer-Verlag, 6 (2008), 339–362. doi: 10.1007/978-0-387-75217-4_8. [10] V. Isakov, Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, Discr. Cont. Dyn. Syst. S, 4 (2011), 631-640.  doi: 10.3934/dcdss.2011.4.631. [11] V. Isakov, Increasing stability of the continuation for general elliptic equations of second order, in New Prospects in Direct, Inverse, and Control Problems for Evolution Equations, Springer INdAM Series 10 (2014), 203–218. doi: 10.1007/978-3-319-11406-4_10. [12] V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-51658-5. [13] V. Isakov and S. Kindermann, Regions of stability in the Cauchy problem for the Helmholtz equation, Methods Appl. of Anal., 18 (2011), 1-29.  doi: 10.4310/MAA.2011.v18.n1.a1. [14] V. Isakov and S. Lu, Inverse source problems without (pseudo)convexity assumptions, Inv. Probl. Imag., 12 (2018), 955-970.  doi: 10.3934/ipi.2018040. [15] V. Isakov and S. Lu, Increasing stability in the inverse source problems with attenuation and many frequencies, SIAM J. Appl. Math., 78 (2018), 1-18.  doi: 10.1137/17M1112704. [16] V. Isakov, R.-Y. Lai and J.-N. Wang, Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594.  doi: 10.1137/15M1019052. [17] V. Isakov and J.-N. Wang, Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to Neumann map, Inv. Probl. Imag., 8 (2014), 1139-1150.  doi: 10.3934/ipi.2014.8.1139. [18] F. John, Continuous Dependence on Data for Solutions of Partial Differential Equations With a Prescribed Bound, Comm. Pure Appl. Math., 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402. [19] L. Robbiano, Theoreme d'unicite adapte au controle des solutions des problemes hyperboliques, Comm. Part. Diff. Equat., 16 (1991), 789-800.  doi: 10.1080/03605309108820778. [20] L. Robbiano, Fonction de cout et controle des solutions des equations hyperboliques, Asympt. Anal., 10 (1995), 95-115. [21] L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients, Invent. Math., 131 (1998), 493-529.  doi: 10.1007/s002220050212. [22] D. Tataru, Unique continuation for solutions to PDE: Between Hörmander's Theorem and Holmgren's Theorem, Comm. Part. Diff. Equat., 20 (1995), 855-884.  doi: 10.1080/03605309508821117. [23] S. Vessella, Stability estimates for an inverse hyperbolic initial boundary value problem with unknown boundaries, SIAM J. Math. Anal., 47 (2016), 1419-1457.  doi: 10.1137/140976212. [24] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.
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