On the stability of the solutions of inverse problems for elliptic equations

Alexander V. Velisevich ; Anna Sh. Lyubanova

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 3, pp. 398-407.

Résumé

The inverse problems on finding the unknown lower coefficient in linear and nonlinear second-order elliptic equations with integral overdetermination conditions are considered. The conditions of overdetermination are given on the boundary of the domain. The continuous dependence of the strong solution on the input data of the inverse problem for the linear equation is proved in the case of the mixed boundary condition. As to the nonlinear equation, the continuous dependence of the strong solution on the overdetermination data is established for the inverse problem with the Dirichlet boundary condition.

Détail

Keywords: inverse problem, integral overdetermination, continuous dependence on input data.
Mots-clés : elliptic equation

Bibliographie

[1] A.V.Velisevich, “On an inverse problem for the stationary equation with a boundary condition of the third type”, J. Siberian Federal University. Mathematics and Physics, 14:5 (2021), 659–666 | DOI | MR | Zbl

[2] A.Sh.Lyubanova, A.V.Velisevich, “An inverse problem fot the quasilinear elliptic equation”, Journal of Mathematical Sciences, 270 (2023), 06370–9 | DOI | MR

[3] A.Sh.Lyubanova, “Identification of a constant coefficient in an elliptic equation”, Applicable Analysis, 87 (2008), 1121–1128 | DOI | MR | Zbl

[4] A.Sh.Lyubanova, A.Tani, “An inverse problem for pseudoparabolic equation of filtration: the existence, uniqueness and regularity”, Applicable Analysis, 90 (2011), 1557–1571 | DOI | MR | Zbl

[5] A.I.Prilepko, D.G.Orlovsky, I.A.Vasin, Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, 2000 | MR | Zbl

[6] A.N.Konovalov, Problems of the filtration of incompressible fluid, Nauka, Novosibirsk, 1988 (in Russian) | MR

[7] R.Helmig, A.Weiss, B.I.Wohlmuth, “Dynamic capillary effects in heterogeneous porous media”, Computational Geosciences, 11:3 (2007), 9050–1 | DOI | MR

[8] G.V.Alekseev, E.A.Kalinina, “Identification of the lower coefficient for the stationary convection-diffusion-reaction equation”, Sib. Zh. Ind. Math., 10:1(29) (2007), 3–16 (in Russian) | MR | Zbl

[9] M.M.Lavrentiev, “On the Cauchy problem for the Laplace equation”, Iz. AN URSS: S. Math., 20:6 (1956), 816–842 (in Russian) | MR

[10] M.M.Lavrentiev, On some ill-posed problems of mathematical physics, Nauka, Novosibirsk, 1962 (in Russian) | MR

[11] M.M.Lavrebtiev, V.G.Romanov, V.G.Vasiliev, Multidimensional inverse problems for differential equations, Nauka, Novosibirsk, 1969 (in Russian) | MR

[12] G.V.Alekseev, V.A.Levin, “Optimization method in problems of thermal masking of material bodies”, Dokl. RAN, 471:1 (2016), 32–36 (in Russian) | DOI | MR

[13] R.A.Aliev, “On some inverse problem for quasilinear equation of the elliptic type”, Iz. Sar. Univ. N. Ser. Math. Mech. Inf., 11:1 (2011), 3–9 (in Russian) | DOI

[14] S.G.Pyatkov, “On some inverse problems for the elliptic equations and systems”, J. Appl. Industr. Math., 5:3 (2010), 417–430 | DOI | MR

[15] G.Alessandrini, R.Caburro, “The Local Calderon Problem and the Determination at the Boundary of the Conductivity”, Comm. Partial Differential Equations, 34 (2009), 918–936 | DOI | MR | Zbl

[16] A.P.Calderon, “On an inverse boundary value problem”, Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, Brazil), Soc. Brazil. Mat., Rio de Janeiro, 1980, 65–73 | MR

[17] F.Kanca, “Inverse coefficient problem for a second-order elliptic equation with nonlocal boundary conditions”, Mathematical Methods in the Applied sciences, 39 (2016), 3152–3158 | DOI | MR | Zbl

[18] M.V.Klibanov, A.Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, the Netherlands, 2004 | MR | Zbl

[19] A.I.Kozhanov, “Nonlinear inverse problems for elliptic equations”, Journal of Inverse and Ill-Posed Problems, 9:4 (2001), 413–424 | DOI | MR | Zbl

[20] M.A.Pilant, “Uniqueness Theorem for Determining Conductivity from Overspecified Boundary Data”, Journal of Mathematical Analysis and Applications, 136:1 (1988), 20–28 | DOI | MR | Zbl

[21] V.V.Soloviov, “Inverse problems for the elliptic equations in space”, Diff. Ur., 47:4 (2011), 499–506 (in Russian) | DOI | MR

[22] M.Franca, A.Sfecci, “On a diffusion model with absorption and production”, Nonlinear Analysis: Real World Applications, 34:1 (2017), 41–60 | DOI | MR | Zbl

[23] Y.T.Mehraliyev, F.Kanca, “An Inverse Boundary Value Problem for a Second Order Elliptic Equation in a Rectangle”, Mathematical Modelling and Analysis, 19:2 (2014), 241–256 | DOI | MR | Zbl

[24] Academic press, New York, 1973

[25] J.-L.Lions, E.Magenes, Non-homogeneous boundary value problems and applications, Dunod, Paris, 1968 | MR