Abstract
We consider the following perturbed polyharmonic operator \(\mathcal {L}(x,D)\) of order 2m defined in a bounded domain \(\Omega \subset \mathbb {R}^n, n\ge 3\) with smooth boundary, as \( \mathcal {L}(x,D) \equiv (-\Delta )^m + \sum _{j,k=1}^{n}A_{jk} D_{j}D_{k} + \sum _{j=1}^{n}B_{j} D_{j} + q(x),\) where A is a symmetric 2-tensor field, B and q are vector field and scalar potential respectively. We show that the coefficients \(A=[A_{jk}]\), \(B=(B_j)\) and q can be recovered from the associated Dirichlet-to-Neumann data on the boundary. Note that this result shows an example of determining higher order (2nd order) symmetric tensor field in the class of inverse boundary value problem.
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Agmon, S.: Lectures on elliptic boundary value problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton (1965)
Anupam, P.C., Horst, H.: Stability of the inverse boundary value problem for the biharmonic operator: logarithmic estimates. J. Inverse Ill-Posed Probl. 25(2), 251–263 (2017)
Ashbaugh, M.S.: On universal inequalities for the low eigenvalues of the buckling problem. In Partial Differential Equations and Inverse Problems. Contemp. Math., vol. 362, pp. 13–31. Am. Math. Soc., Providence, RI (2004)
Assylbekov, Y.M., Iyer, K.: Determining rough first order perturbations of the polyharmonic operator. (2017) arXiv:1703.02569
Assylbekov, Y.M.: Inverse problems for the perturbed polyharmonic operator with coefficients in Sobolev spaces with non-positive order. Inverse Probl. 32(10), 105009 (2016)
Assylbekov, Y.M., Yang, Y.: Determining the first order perturbation of a polyharmonic operator on admissible manifolds. J. Differ. Equ. 262(1), 590–614 (2017)
Bhattacharyya, S.: An inverse problem for the magnetic schrödinger operator on riemannian manifolds from partial boundary data. Inverse Probl. Imaging 12, 801–830 (2018)
Bukhgeim, A., Uhlmann, G.: Recovering a potential from cauchy data. Commun. Partial Differ. Equ. 27(3–4), 653–688 (2007)
Calderón, A.-P.: On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), pp. 65–73. Soc. Brasil. Mat., Rio de Janeiro (1980)
Choudhury, A.P., Krishnan, V.P.: Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials. J. Math. Anal. Appl. 431(1), 300–316 (2015)
Dos Santos, D., Ferreira, C.E., Kenig, J.S., Uhlmann, G.: Determining a magnetic Schrödinger operator from partial Cauchy data. Commun. Math. Phys. 271(2), 467–488 (2007)
Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic boundary value problems. In: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains. Lecture Notes in Mathematics, vol. 1991. Springer, Berlin (2010)
Ghosh, T.: An inverse problem on determining upto first order perturbations of a fourth order operator with partial boundary data. Inverse Probl. 31(10), 105009 (2015)
Ghosh, T., Krishnan, V.: Determination of lower order perturbations of the polyharmonic operator from partial boundary data. Appl. Anal. 94, 2444–2463 (2015)
Grubb, G.: Distributions and Operators. Graduate Texts in Mathematics, vol. 252. Springer, New York (2009)
Ikehata, M.: A special Green’s function for the biharmonic operator and its application to an inverse boundary value problem. Comput. Math. Appl. 22(4–5), 53–66 (1991). (Multidimensional inverse problems)
Isakov, V.: Completeness of products of solutions and some inverse problems for PDE. J. Differ. Equ. 92(2), 305–316 (1991)
Kenig, C.E., Sjöstrand, J., Uhlmann, G.: The Calderón problem with partial data. Ann. Math. (2) 165(2), 567–591 (2007)
Krupchyk, K., Uhlmann, G.: Inverse boundary problems for polyharmonic operators with unbounded potentials. J. Spectr. Theory 6(1), 145–183 (2016)
Krupchyk, K., Lassas, M., Uhlmann, G.: Determining a first order perturbation of the biharmonic operator by partial boundary measurements. J. Funct. Anal. 262(4), 1781–1801 (2012)
Krupchyk, K., Lassas, M., Uhlmann, G.: Inverse boundary value problems for the perturbed polyharmonic operator. Trans. Am. Math. Soc. 366(1), 95–112 (2014)
Salo, M.: Inverse problems for nonsmooth first order perturbations of the Laplacian. Ann. Acad. Sci. Fenn. Math. Diss. (139): 67. Dissertation, University of Helsinki, Helsinki (2004)
Serov, V.S.: Borg-Levinson theorem for perturbations of the bi-harmonic operator. Inverse Probl. 32(4), 045002 (2016)
Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. Inverse and Ill-Posed Problems Series. VSP, Utrecht (1994)
Sharafutdinov, V.A.: Integral Geometry of Tensor Fields. VSP, Utrecht, the Netherlands (1994)
Sun, Z.Q.: An inverse boundary value problem for Schrödinger operators with vector potentials. Trans. Am. Math. Soc. 338(2), 953–969 (1993)
Sylvester, J., Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. Math. (2) 125(1), 153–169 (1987)
Tyni, T., Serov, V.: Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Probl. Imaging 12, 205–227 (2018)
Yang, Y.: Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data. J. Differ. Equ. 257(10), 3607–3639 (2014)
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The authors express their gratitude to the anonymous referees for their valuable comments and suggestions on this article. Bhattacharyya was partially supported by the Postdoctoral fellowship from Institute for Advanced Study, The Hong Kong University of Science and Technology.
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Communicated by Eric Todd Quinto.
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Bhattacharyya, S., Ghosh, T. Inverse Boundary Value Problem of Determining Up to a Second Order Tensor Appear in the Lower Order Perturbation of a Polyharmonic Operator. J Fourier Anal Appl 25, 661–683 (2019). https://doi.org/10.1007/s00041-018-9625-3
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DOI: https://doi.org/10.1007/s00041-018-9625-3