Solvability of interior transmission problem for the diffusion equation by constructing its Green function

  • 1 Department of Mathematics, Hokkaido University, 060-0810, Sapporo, Japan
  • 2 School of Mathematics, Southeast University, 210096, Nanjing, P. R. China
Gen Nakamura and Haibing WangORCID iD: https://orcid.org/0000-0001-9288-3856

Abstract

Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove the unique solvability of the interior transmission problem by constructing its Green function. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the local parametrix in the Laplace domain and estimating it there, a local parametrix for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to generate a global parametrix for the parabolic interior transmission problem and then compensate it to get the Green function by the Levi method. The uniqueness of the Green function is justified by using the duality argument, and then the unique solvability of the interior transmission problem is concluded. We would like to emphasize that the Green function for the parabolic interior transmission problem is constructed for the first time in this paper. It can be applied for active thermography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.

  • [1]

    R. Arima, On general boundary value problem for parabolic equations, J. Math. Kyoto Univ. 4 (1964), 207–243.

    • Crossref
    • Export Citation
  • [2]

    F. Cakoni, D. Colton and D. Gintides, The interior transmission eigenvalue problem, SIAM J. Math. Anal. 42 (2010), no. 6, 2912–2921.

    • Crossref
    • Export Citation
  • [3]

    F. Cakoni, D. Colton and H. Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Anal. 42 (2010), no. 1, 145–162.

    • Crossref
    • Export Citation
  • [4]

    F. Cakoni, D. Colton and H. Haddar, Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF Regional Conf. Ser. in Appl. Math. 88, Society for Industrial and Applied Mathematics, Philadelphia, 2016.

  • [5]

    F. Cakoni, D. Gintides and H. Haddar, The existence of an infinite discrete set of transmission eigenvalues, SIAM J. Math. Anal. 42 (2010), no. 1, 237–255.

    • Crossref
    • Export Citation
  • [6]

    D. Colton, L. Päivärinta and J. Sylvester, The interior transmission problem, Inverse Probl. Imaging 1 (2007), no. 1, 13–28.

    • Crossref
    • Export Citation
  • [7]

    A. Cossonnière and H. Haddar, The electromagnetic interior transmission problem for regions with cavities, SIAM J. Math. Anal. 43 (2011), no. 4, 1698–1715.

    • Crossref
    • Export Citation
  • [8]

    M. Faierman, The interior transmission problem: Spectral theory, SIAM J. Math. Anal. 46 (2014), no. 1, 803–819.

    • Crossref
    • Export Citation
  • [9]

    A. García, E. V. Vesalainen and M. Zubeldia, Discreteness of transmission eigenvalues for higher-order main terms and perturbations, SIAM J. Math. Anal. 48 (2016), no. 4, 2382–2398.

    • Crossref
    • Export Citation
  • [10]

    P. Greiner, An asymptotic expansion for the heat equation, Arch. Ration. Mech. Anal. 41 (1971), 163–218.

    • Crossref
    • Export Citation
  • [11]

    M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for operators with constant coefficients, SIAM J. Math. Anal. 42 (2010), no. 6, 2965–2986.

    • Crossref
    • Export Citation
  • [12]

    M. Hitrik, K. Krupchyk, P. Ola and L. Päivärinta, Transmission eigenvalues for elliptic operators, SIAM J. Math. Anal. 43 (2011), no. 6, 2630–2639.

    • Crossref
    • Export Citation
  • [13]

    V. Isakov, K. Kim and G. Nakamura, Reconstruction of an unknown inclusion by thermography, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 4, 725–758.

  • [14]

    S. Itô, Diffusion Equations, Transl. Math. Monogr. 114, American Mathematical Society, Providence, 1992.

  • [15]

    X. Ji and H. Liu, On isotropic cloaking and interior transmission eigenvalue problems, European J. Appl. Math. 29 (2018), no. 2, 253–280.

    • Crossref
    • Export Citation
  • [16]

    K. Kim, G. Nakamura and M. Sini, The Green function of the interior transmission problem and its applications, Inverse Probl. Imaging 6 (2012), no. 3, 487–521.

    • Crossref
    • Export Citation
  • [17]

    A. Kirsch, The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math. 37 (1986), no. 3, 213–225.

    • Crossref
    • Export Citation
  • [18]

    A. Kirsch, On the existence of transmission eigenvalues, Inverse Probl. Imaging 3 (2009), no. 2, 155–172.

    • Crossref
    • Export Citation
  • [19]

    E. Lakshtanov and B. Vainberg, Ellipticity in the interior transmission problem in anisotropic media, SIAM J. Math. Anal. 44 (2012), no. 2, 1165–1174.

    • Crossref
    • Export Citation
  • [20]

    E. Lakshtanov and B. Vainberg, Sharp Weyl law for signed counting function of positive interior transmission eigenvalues, SIAM J. Math. Anal. 47 (2015), no. 4, 3212–3234.

    • Crossref
    • Export Citation
  • [21]

    A. Lechleiter and M. Rennoch, Inside-outside duality and the determination of electromagnetic interior transmission eigenvalues, SIAM J. Math. Anal. 47 (2015), no. 1, 684–705.

    • Crossref
    • Export Citation
  • [22]

    X. Li, J. Li, H. Liu and Y. Wang, Electromagnetic interior transmission eigenvalue problem for inhomogeneous media containing obstacles and its applications to near cloaking, IMA J. Appl. Math. 82 (2017), 1013–1042.

    • Crossref
    • Export Citation
  • [23]

    P. Monk and J. Sun, Finite element methods for Maxwell’s transmission eigenvalues, SIAM J. Sci. Comput. 34 (2012), no. 3, B247–B264.

    • Crossref
    • Export Citation
  • [24]

    G. Nakamura and H. Wang, Linear sampling method for the heat equation with inclusions, Inverse Problems 29 (2013), no. 10, Article ID 104015.

  • [25]

    G. Nakamura and H. Wang, Reconstruction of an unknown cavity with Robin boundary condition inside a heat conductor, Inverse Problems 31 (2015), no. 12, Article ID 125001.

  • [26]

    G. Nakamura and H. Wang, Numerical reconstruction of unknown Robin inclusions inside a heat conductor by a non-iterative method, Inverse Problems 33 (2017), no. 5, Article ID 055002.

  • [27]

    L. Päivärinta and J. Sylvester, Transmission eigenvalues, SIAM J. Math. Anal. 40 (2008), no. 2, 738–753.

    • Crossref
    • Export Citation
  • [28]

    B. P. Rynne and B. D. Sleeman, The interior transmission problem and inverse scattering from inhomogeneous media, SIAM J. Math. Anal. 22 (1991), no. 6, 1755–1762.

    • Crossref
    • Export Citation
  • [29]

    R. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969), 889–920.

    • Crossref
    • Export Citation
  • [30]

    M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer, Berlin, 1987.

  • [31]

    J. Sun, Iterative methods for transmission eigenvalues, SIAM J. Numer. Anal. 49 (2011), no. 5, 1860–1874.

    • Crossref
    • Export Citation
  • [32]

    J. Sylvester, Discreteness of transmission eigenvalues via upper triangular compact operators, SIAM J. Math. Anal. 44 (2012), no. 1, 341–354.

    • Crossref
    • Export Citation
  • [33]

    J. Sylvester, Transmission eigenvalues in one dimension, Inverse Problems 29 (2013), no. 10, Article ID 104009.

  • [34]

    H. Wang and Y. Li, Numerical solution of an inverse boundary value problem for the heat equation with unknown inclusions, J. Comput. Phys. 369 (2018), 1–15.

    • Crossref
    • Export Citation
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