Inverse source and inverse coefficients problems for elliptic and parabolic equations in Hölder and Sobolev spaces

A. I. Prilepko   ; A. B. Kostin   ; V. V. Solov'ev

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 17 (2017) no. 3, pp. 67-85

Résumé

We give a review of some results obtained during the last fifteen years. This results includes the theorems of existence and uniqueness of the solution of linear and nonlinear inverse problems of reconstruction of unknown coefficients in elliptic and parabolic equations.

Détail

Keywords: inverse coefficients problems
Mots-clés : elliptic equations, parabolic equations, final observation, nonlocal observation.

Bibliographie

[1] A. V. Bitsadze, Boundary Value Problems for Second Order Elliptic Equations, Elsevier, 2012 | MR

[2] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, N.Y., 2000 | MR | Zbl

[3] A. I. Prilepko, “The semigroup method for inverse, nonlocal, and nonclassical problems. prediction-control and prediction-observation for evolution equations: I”, Differ. Equ., 41:11 (2005), 1635–1646 | DOI | MR | Zbl

[4] A. B. Kostin, “Basis property of a system of functions related to the inverse problem of finding the source”, Differ. Equ., 44:2 (2008), 256–266 | DOI | MR | Zbl

[5] V. L. Kamynin, A. B. Kostin, “Two inverse problems of finding a coefficient in a parabolic equation”, Differ. Equ., 46:3 (2010), 375–386 | DOI | MR | Zbl

[6] A. B. Kostin, “The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation”, Mat. Sb., 204:10 (2013), 1391 | DOI | DOI | MR | Zbl

[7] A. B. Kostin, “Complex eigenvalues of elliptic operators and an example of inverse problem with nonunique solution”, Dokl. Math., 88:3 (2013), 651–654 | DOI | DOI | MR | Zbl

[8] V. L. Kamynin, A. B. Kostin, “Inverse problem of finding $n$ coefficients of lower derivatives in a parabolic equation”, Differ. Equ., 50:4 (2014), 476–488 | DOI | DOI | MR | Zbl

[9] A. B. Kostin, “Counterexamples in inverse problems for parabolic, elliptic, and hyperbolic equations”, Comp. Math. and Math. Phys., 54:5 (2014), 797–810 | DOI | DOI | MR | Zbl

[10] A. B. Kostin, “Inverse problem of finding the coefficient of $\mathbf{u}$ in a parabolic equation on the basis of a nonlocal observation condition”, Differ. Equ., 51:5 (2015), 605–619 | DOI | DOI | MR | Zbl

[11] V. V. Solov'ev, “Inverse problems for elliptic equations in the space: II”, Differ. Equ., 47:5 (2011), 715–725 | DOI | MR | Zbl

[12] V. V. Solov'ev, “Inverse coefficient problems for elliptic equations in a cylinder: I”, Differ. Equ., 49:8 (2013), 996–1005 | DOI | MR | Zbl

[13] V. V. Solov'ev, “Inverse coefficient problems for elliptic equations in a cylinder: II”, Differ. Equ., 49:12 (2013), 1560–1568 | DOI | MR | Zbl

[14] M. M. Lavrent'ev, V. G. Romanov, V. G. Vasil'ev, Multidimensional Inverse Problems for Differential Equations, Lecture Notes in Mathematics, 167, Springer Verlag, Berlin, 1974 | MR

[15] A. D. Iskenderov, R. G. Tagiev, “The inverse problem of determining the right-hand sides of evolution equations in Banach space”, Nauchn. Trudy Azerbaidzhan. Gos. Univ., 1 (1979), 51–56 | Zbl | Zbl

[16] Rundell W., “An Inverse Problem for an Elliptic Partial Differential Equation”, J. Math. Anall. Appl., 126 (1987), 329–340 | DOI | MR | Zbl

[17] D. G. Orlovskii, “On one inverse problem for differential equation of the second order in Banach space”, Differ. Uravn., 25:6 (1989), 1000–1009 (in Russian)

[18] D. G. Orlovskii, “On a problem of determining the parameter of an evolution equation”, Differ. Equ., 26:9 (1990), 1201–1207 | MR

[19] A. I. Prilepko, “Selected questions in inverse problems of mathematical physics”, Uslovno-korrektnye zadachi matematicheskoi fiziki i analiza, 1992, 51–63 (in Russian)

[20] A. Khaidarov, “A class of inverse problems for elliptic equations”, Differ. Equ., 23:8 (1987), 939–945 | MR | Zbl

[21] A. Khaidarov, “A class of inverse problems for elliptic equations”, Sib. Mat. Zh., 31:4 (1990), 149–159 (in Russian) | Zbl

[22] Y. S. Eidel'man, “Some inverse problems for parabolic and elliptic differential-operator equations”, Ukr. Math. J., 45:1 (1993), 132–140 | MR | Zbl

[23] O. Y. Imanuvilov, “Some class of inverse problems for semilinear elliptic and parabolic equations”, Trudy Moskov. Mat. Obshch., 55, 1994, 285–309 (in Russian) | Zbl

[24] Isakov V. M., Inverse Problems for Partial Differential Equations, Springer, N.Y., 2006 | MR | Zbl

[25] A. I. Prilepko, A. B. Kostin, “On certain inverse problems for parabolic equations with final and integral observation”, Mat. Sb., 183:4 (1992), 49–68 (in Russian) | Zbl

[26] A. I. Prilepko, A. B. Kostin, “Inverse problems of the determination of the coefficient in parabolic equations. I”, Sib. Math. J., 33:3 (1992), 489–496 | DOI | MR | Zbl

[27] A. I. Prilepko, A. B. Kostin, “Inverse problems of the determination of the coefficient in parabolic equations. II”, Sib. Math. J., 34:5 (1993), 923–937 | DOI | MR | Zbl

[28] A. I. Prilepko, V. V. Soloviev, “Solvability theorems and Rotes method for inverse problems for a parabolic equation. I”, Differ. Equ., 23:10 (1987), 1230–1237 | MR | Zbl | Zbl

[29] V. V. Solov'ev, “Solvability of the inverse problem of finding a source, using overdetermination on the upper base for a parabolic equation”, Differ. Equ., 25:9 (1989), 1114–1119 | Zbl

[30] Isakov V. M., “Inverse Parabolic Problems with the Final Overdetermination”, Comm. on Pure and Appl. Math., 44 (1991), 185–209 | DOI | MR | Zbl

[31] A. I. Prilepko, D. S. Tkachenko, “Properties of solutions to the parabolic equation and uniqueness of the solution to the inverse source problem with integral overdetermination”, Comp. Math. and Math. Phys., 43:4 (2003), 537–546 | MR | Zbl

[32] A. I. Prilepko, D. S. Tkachenko, “Fredholm property of the inverse source problem for parabolic systems”, Differ. Equ., 39:12 (2003), 1785–1793 | DOI | MR | Zbl

[33] A. I. Prilepko, I. V. Tikhonov, “Recovery of the nonhomogeneous term in an abstract evolution equation”, Izv. Math., 44:2 (1995), 373–394 | DOI | MR | Zbl

[34] Rundell W., “Determination of an Unknown Non-Homogeneous Term in a Linear Partial Differential Equation from over Specified Boundary Data”, Appl. Anal., 10:10 (1980), 231–240 | DOI | MR

[35] Y. S. Eidel'man, “Uniqueness of solution an inverse problem for differential equation in Banach space”, Differ. Uravn., 23:9 (1987), 1647–1649 (in Russian) | Zbl

[36] A. I. Prilepko, I. V. Tikhonov, “The principle of the positity of a solution to a linear inverse problem and its application to the heat conduction coefficient problem”, Dokl. Akad. Nauk, 364:1 (1999), 21–23 (in Russian) | Zbl

[37] I. V. Tikhonov, Yu. S. Eidel'man, “Uniqueness criterion in an inverse problem for an abstract differential equation with nonstationary inhomogeneous term”, Math. Notes, 77:2 (2005), 246–262 | DOI | DOI | MR | Zbl

[38] V. L. Kamynin, “On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition”, Math. Notes, 73:1–2 (2003), 202–211 | DOI | DOI | MR | Zbl

[39] V. L. Kamynin, “On the inverse problem of determining the right-hand side of a parabolic equation under an integral overdetermination condition”, Math. Notes, 77:3–4 (2005), 482–493 | DOI | DOI | MR | Zbl

[40] A. B. Kostin, “Solvability of one moments problem and its relationship with parabolic inverse problem”, Vestn. of Moscow Univ., 1995, no. 1, 28–33 (in Russian)

[41] A. I. Prilepko, V. V. Soloviev, “On the solvability of inverse boundary value problems for the determination of the coefficient preceding the lower derivative in a parabolic equation”, Differ. Equ., 23:1 (1987), 101–107 | MR | Zbl

[42] A. I. Kozhanov, “Nonlinear loaded equations and inverse problems”, Zh. Vychisl. Mat. Mat. Fiz., 44:4 (2004), 694–716 (in Russian) | Zbl

[43] A. I. Kozhanov, “Solvability of the inverse problem of finding thermal conductivity”, Sib. Math. J., 46:5 (2005), 841–856 | DOI | MR | Zbl

[44] A. I. Kozhanov, “Parabolic equations with an unknown absorption coefficient”, Dokl. Math., 74:1 (2006), 573–576 | DOI | MR | Zbl

[45] S. G. Pyatkov, B. N. Tsybikov, “Some classes of inverse evolution problems for parabolic equations”, Sib. Math. J., 50:1 (2009), 141–153 | DOI | MR | Zbl

[46] Kozhanov A. I., Safiullova R. R., “Linear Inverse Problems for Parabolic and Hyperbolic Equations”, J. of Inv. and Ill-Posed Probl., 18:1 (2010), 1–24 | DOI | MR | Zbl

[47] V. L. Kamynin, “Inverse problem of finding the coefficient of a lower derivative in a parabolic equation on the plane”, Differ. Equ., 48:2 (2012), 214–223 | DOI | MR | Zbl

[48] V. L. Kamynin, “The inverse problem of determining the lower-order coefficient in parabolic equations with integral observation”, Math. Notes, 94:1–2 (2013), 205–213 | DOI | DOI | MR | Zbl