\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Identifiability of diffusion coefficients for source terms of non-uniform sign

  • * Corresponding author

    * Corresponding author 

The authors acknowledge support by the Hausdorff Center of Mathematics, University of Bonn

Abstract Full Text(HTML) Related Papers Cited by
  • The problem of recovering a diffusion coefficient $ a $ in a second-order elliptic partial differential equation from a corresponding solution $ u $ for a given right-hand side $ f $ is considered, with particular focus on the case where $ f $ is allowed to take both positive and negative values. Identifiability of $ a $ from $ u $ is shown under mild smoothness requirements on $ a $, $ f $, and on the spatial domain $ D $, assuming that either the gradient of $ u $ is nonzero almost everywhere, or that $ f $ as a distribution does not vanish on any open subset of $ D $. Further results of this type under essentially minimal regularity conditions are obtained for the case of $ D $ being an interval, including detailed information on the continuity properties of the mapping from $ u $ to $ a $.

    Mathematics Subject Classification: 35R30, 35J25.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] G. Alessandrini, On the identification of the leading coefficient of an elliptic equation, Boll. Un. Mat. Ital. C (6), 4 (1985), 87-111. 
    [2] G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl. (4), 145 (1986), 265-295.  doi: 10.1007/BF01790543.
    [3] A. BonitoA. CohenR. DeVoreG. Petrova and G. Welper, Diffusion coefficients estimation for elliptic partial differential equations, SIAM J. Math. Anal., 49 (2017), 1570-1592.  doi: 10.1137/16M1094476.
    [4] G. Chavent and K. Kunisch, The output least squares identifiability of the diffusion coefficient from an $H^1$-observation in a 2-D elliptic equation, A tribute to J. L. Lions, ESAIM Control Optim. Calc. Var., 8 (2002), 423-440.  doi: 10.1051/cocv:2002028.
    [5] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Rational Mech. Anal., 147 (1999), 89-118.  doi: 10.1007/s002050050146.
    [6] G.-Q. ChenM. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Communications on Pure and Applied Mathematics, 62 (2009), 242-304.  doi: 10.1002/cpa.20262.
    [7] C. Chicone and J. Gerlach, A note on the identifiability of distributed parameters in elliptic equations, SIAM J. Math. Anal., 18 (1987), 1378-1384.  doi: 10.1137/0518099.
    [8] R. S. Falk, Error estimates for the numerical identification of a variable coefficient, Math. Comp., 40 (1983), 537-546.  doi: 10.1090/S0025-5718-1983-0689469-3.
    [9] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.
    [10] N. Honda, J. McLaughlin and G. Nakamura, Conditional stability for a single interior measurement, Inverse Problems, 30 (2014), 055001l, 19pp. doi: 10.1088/0266-5611/30/5/055001.
    [11] K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems, J. Math. Anal. Appl., 188 (1994), 1040-1066.  doi: 10.1006/jmaa.1994.1479.
    [12] R. V. Kohn and B. D. Lowe, A variational method for parameter identification, RAIRO Modél. Math. Anal. Numér., 22 (1988), 119-158.  doi: 10.1051/m2an/1988220101191.
    [13] K. Kunisch, Inherent identifiability of parameters in elliptic differential equations, J. Math. Anal. Appl., 132 (1988), 453-472.  doi: 10.1016/0022-247X(88)90074-1.
    [14] K. Kunisch and L. W. White, Identifiability under approximation for an elliptic boundary value problem, SIAM J. Control Optim., 25 (1987), 279-297.  doi: 10.1137/0325017.
    [15] F. MaggiSets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge University Press, 2012.  doi: 10.1017/CBO9781139108133.
    [16] P. Marcellini, Identificazione di un coefficiente in una equazione differenziale ordinaria del secondo ordine, Ricerche Mat, 31 (1982), 223-243. 
    [17] G. R. Richter, An inverse problem for the steady state diffusion equation, SIAM J. Appl. Math., 41 (1981), 210-221.  doi: 10.1137/0141016.
    [18] G. R. Richter, Numerical identification of a spatially varying diffusion coefficient, Math. Comp., 36 (1981), 375-386.  doi: 10.1090/S0025-5718-1981-0606502-3.
    [19] V. Volterra, Sui Principii del Calcolo Integrale, Giornale di Matematiche, 1881.
  • 加载中
SHARE

Article Metrics

HTML views(1337) PDF downloads(239) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return