## Duke Mathematical Journal

### Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets

#### Abstract

We consider the inverse problem to determine a smooth compact Riemannian manifold with boundary $(M,g)$ from a restriction $\Lambda_{\mathcal{S},\mathcal{R}}$ of the Dirichlet-to-Neumann operator for the wave equation on the manifold. Here $\mathcal{S}$ and $\mathcal{R}$ are open sets in $\partial M$ and the restriction $\Lambda_{\mathcal{S},\mathcal{R}}$ corresponds to the case where the Dirichlet data is supported on $\mathbb{R}_{+}\times\mathcal{S}$ and the Neumann data is measured on $\mathbb{R}_{+}\times\mathcal{R}$. In the novel case where $\overline{\mathcal{S}}\cap\overline{\mathcal{R}}=\emptyset$, we show that $\Lambda_{\mathcal{S},\mathcal{R}}$ determines the manifold $(M,g)$ uniquely, assuming that the wave equation is exactly controllable from the set of sources $\mathcal{S}$. Moreover, we show that the exact controllability can be replaced by the Hassell–Tao condition for eigenvalues and eigenfunctions, that is,

$$\lambda_{j}\le C\Vert \partial _{\nu}\phi_{j}\Vert _{L^{2}(\mathcal{S})}^{2},\quad j=1,2,\dots,$$

where $\lambda_{j}$ are the Dirichlet eigenvalues and where $(\phi_{j})_{j=1}^{\infty}$ is an orthonormal basis of the corresponding eigenfunctions.

#### Article information

Source
Duke Math. J., Volume 163, Number 6 (2014), 1071-1103.

Dates
First available in Project Euclid: 11 April 2014

https://projecteuclid.org/euclid.dmj/1397223296

Digital Object Identifier
doi:10.1215/00127094-2649534

Mathematical Reviews number (MathSciNet)
MR3192525

Zentralblatt MATH identifier
1375.35634

Subjects
Primary: 35R30: Inverse problems
Secondary: 35R01: Partial differential equations on manifolds [See also 32Wxx, 53Cxx, 58Jxx]

#### Citation

Lassas, Matti; Oksanen, Lauri. Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets. Duke Math. J. 163 (2014), no. 6, 1071--1103. doi:10.1215/00127094-2649534. https://projecteuclid.org/euclid.dmj/1397223296

#### References

• [1] R. Alexander and S. Alexander, Geodesics in Riemannian manifolds-with-boundary, Indiana Univ. Math. J. 30 (1981), 481–488.
• [2] M. Anderson, A. Katsuda, Y. Kurylev, M. Lassas, and M. Taylor, Boundary regularity for the Ricci equation, geometric convergence, and Gel’fand’s inverse boundary problem, Invent. Math. 158 (2004), 261–321.
• [3] K. Astala and L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math. (2) 163 (2006), 265–299.
• [4] K. Astala, L. Päivärinta, and M. Lassas, Calderón’s inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations 30 (2005), 207–224.
• [5] C. Bardos, G. Lebeau, and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim. 30 (1992), 1024–1065.
• [6] M. I. Belishev, An approach to multidimensional inverse problems for the wave equation (in Russian), Dokl. Akad. Nauk SSSR 297, no. 3 (1987), 524–527; English translation in Soviet Math. Dokl. 36, no. 3 (1988), 481–484.
• [7] M. I. Belishev, Recent progress in the boundary control method, Inverse Problems 23 (2007), R1–R67.
• [8] M. I. Belishev and Y. Kurylev, To the reconstruction of a Riemannian manifold via its spectral data (BC-method), Comm. Partial Differential Equations 17 (1992), 767–804.
• [9] A. S. Blagoveščenskiĭ, A one-dimensional inverse boundary value problem for a second order hyperbolic equation, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 15 (1969), 85–90.
• [10] A. S. Blagoveščenskiĭ, The inverse boundary value problem of the theory of wave propagation in an anisotropic medium, Trudy Mat. Inst. Steklov. 115 (1971), 39–56. (errata insert), 1971.
• [11] A. L. Bukhgeim, Recovering a potential from Cauchy data in the two-dimensional case, J. Inverse Ill-Posed Probl. 16 (2008), 19–33.
• [12] N. Burq and P. Gérard, Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes, C. R. Math. Acad. Sci. Paris Sér. I 325 (1997), 749–752.
• [13] A.-P. Calderón, “On an inverse boundary value problem” in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Sociedade Brasileira de Matemática, Rio de Janeiro, 1980, 65–73.
• [14] I. Chavel, Riemannian Geometry: A Modern Introduction, 2nd ed., Cambridge Stud. Adv. Math. 98, Cambridge Univ. Press, Cambridge, 2006.
• [15] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math. 178 (2009), 119–171.
• [16] C. Guillarmou and L. Tzou, Calderón inverse problem with partial data on Riemann surfaces, Duke Math. J. 158 (2011), 83–120.
• [17] A. Hassell and T. Tao, Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions, Math. Res. Lett. 9 (2002), 289–305. Correction, Math. Res. Lett. 17 (2010), 793–794.
• [18] T. Helin, M. Lassas, and L. Oksanen, An inverse problem for the wave equation with one measurement and the pseudorandom source, Anal. PDE 5 (2012), 887–912.
• [19] G. Henkin and V. Michel, On the explicit reconstruction of a Riemann surface from its Dirichlet-Neumann operator, Geom. Funct. Anal. 17 (2007), 116–155.
• [20] G. Henkin and V. Michel, Inverse conductivity problem on Riemann surfaces, J. Geom. Anal. 18 (2008), 1033–1052.
• [21] G. Henkin and R. G. Novikov, On the reconstruction of conductivity of a bordered two-dimensional surface in $\mathbb{R}^{3}$ from electrical current measurements on its boundary, J. Geom. Anal. 21 (2011), 543–587.
• [22] G. Henkin and M. Santacesaria, On an inverse problem for anisotropic conductivity in the plane, Inverse Problems 26 (2010), art. ID 095011.
• [23] O. Y. Imanuvilov, G. Uhlmann, and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Amer. Math. Soc. 23 (2010), 655–691.
• [24] O. Y. Imanuvilov, G. Uhlmann, and M. Yamamoto, Determination of second-order elliptic operators in two dimensions from partial Cauchy data, Proc. Natl. Acad. Sci. USA 108 (2011), 467–472.
• [25] O. Y. Imanuvilov, G. Uhlmann, and M. Yamamoto, Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets, Inverse Problems 27 (2011), art. ID 085007.
• [26] H. Isozaki, Y. Kurylev, and M. Lassas, Forward and inverse scattering on manifolds with asymptotically cylindrical ends, J. Funct. Anal. 258 (2010), 2060–2118.
• [27] A. Katchalov and Y. Kurylev, Multidimensional inverse problem with incomplete boundary spectral data, Comm. Partial Differential Equations 23 (1998), 55–95.
• [28] A. Katchalov, Y. Kurylev, and M. Lassas, Inverse Boundary Spectral Problems, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 123, Chapman & Hall/CRC, Boca Raton, Fla., 2001.
• [29] C. E. Kenig, J. Sjöstrand, and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2) 165 (2007), 567–591.
• [30] M. G. Kreĭn, Determination of the density of a nonhomogeneous symmetric cord by its frequency spectrum, Doklady Akad. Nauk SSSR (N.S.) 76 (1951), 345–348.
• [31] I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), 149–192.
• [32] M. Lassas and L. Oksanen, An inverse problem for a wave equation with sources and observations on disjoint sets, Inverse Problems 26 (2010), art. ID 085012.
• [33] M. Lassas, M. Taylor, and G. Uhlmann, The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary, Comm. Anal. Geom. 11 (2003), 207–221.
• [34] M. Lassas and G. Uhlmann, On determining a Riemannian manifold from the Dirichlet-to-Neumann map, Ann. Sci. Éc. Norm. Supér. (4) 34 (2001), 771–787.
• [35] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Grad. Texts in Math. 176, Springer, New York, 1997.
• [36] J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math. 42 (1989), 1097–1112.
• [37] A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2) 143 (1996), 71–96.
• [38] L. Oksanen, Solving an inverse problem for the wave equation by using a minimization algorithm and time-reversed measurements, Inverse Probl. Imaging 5 (2011), 731–744.
• [39] Rakesh, Characterization of transmission data for Webster’s horn equation, Inverse Problems 16 (2000), L9–L24.
• [40] Rakesh and P. Sacks, Uniqueness for a hyperbolic inverse problem with angular control on the coefficients, J. Inverse Ill-Posed Probl. 19 (2011), 107–126.
• [41] P. Stefanov and G. Uhlmann, Integral geometry on tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math. 130 (2008), 239–268.
• [42] J. Sylvester, An anisotropic inverse boundary value problem, Comm. Pure Appl. Math. 43 (1990), 201–232.
• [43] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2) 125 (1987), 153–169.
• [44] D. Tataru, Unique continuation for solutions to PDE’s: Between Hörmander’s theorem and Holmgren’s theorem, Comm. Partial Differential Equations 20 (1995), 855–884.
• [45] D. Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl. (9) 78 (1999), 505–521.
• [46] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1944.