On Carleman-type formulas for solutions to the heat equation

Ilya A. Kurilenko ; Alexander A. Shlapunov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 12 (2019) no. 4, pp. 421-433.

Résumé

We apply the method of integral representations to study the ill-posed Cauchy problem for the heat equation. More precisely we recover a function, satisfying the heat equation in a cylindrical domain, via its values and the values of its normal derivative on a given part of the lateral surface of the cylinder. We prove that the problem is ill-posed in the natural (anisotropic) spaces (Sobolev and Hölder spaces, etc). Finally, we obtain a uniqueness theorem for the problem and a criterion of its solvability and a Carleman-type formula for its solution.

Détail

Keywords: the heat equation, ill-posed problems, integral representation method
Mots-clés : Carleman formulas.

Bibliographie

[1] M.M. Lavrent'ev, “On the Cauchy problem for Laplace equation”, Izvestia AN SSSR. Ser. matem., 1956, no. 20 (in Russian)

[2] M.M. Lavrent'ev, V.G. Romanov, S.P. Shishatskii, Ill-posed problems of mathematical physics and analysis, Nauka, M., 1980 (in Russian) | MR | Zbl

[3] A.N. Tihonov, V.Ya. Arsenin, Methods of solving ill-posed problems, Nauka, M., 1986 | MR

[4] L.A. Aizenberg, Carleman formulas in complex analysis. First applications, Kluwer Ac. Publ., Dordrecht, 1993 | MR

[5] L.A. Aizenberg, A.M. Kytmanov, “On the possibility of holomorphic continuation to a domain of functions given on a part of its boundary”, Mathematics of the USSR-Sbornik, 72:2 (1992), 467–483 | DOI | MR

[6] A.A. Shlyapunov, “On the Cauchy Problem for the Laplace Equation”, Siberian Math. J., 33:3 (1992), 205–215 | MR

[7] A.A. Shlapunov, N. Tarkhanov, “Bases with double orthogonality in the Cauchy problem for systems with injective symbols”, Proc. London. Math. Soc., 71:1 (1995), 1–54 | DOI | MR

[8] N. Tarkhanov, The Cauchy problem for solutions of elliptic equations, Akademie-Verlag, Berlin, 1995 | MR | Zbl

[9] D.P. Fedchenko, A.A. Shlapunov, “On the Cauchy problem for the elliptic complexes in spaces of distributions”, Complex variables and elliptic equations, 59:5 (2014), 651–679 | DOI | MR | Zbl

[10] R.E. Puzyrev, A.A. Shlapunov, “On an ill-Posed problem for the heat equation”, J. Sib. Fed. Univ., Math. and Physics, 5:3 (2012), 337–348

[11] R.E. Puzyrev, A.A. Shlapunov, “On a mixed problem for the parabolic Lamé type operator”, J. Inv. Ill-posed Problems, 23:6 (2015), 555–570 | DOI | MR | Zbl

[12] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva, Linear and quasilinear equations of parabolic type, Nauka, M., 1967 (in Russian) | MR | Zbl

[13] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964 | MR | Zbl

[14] N.V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Math., 12, AMS, Providence, Rhode Island, 1996 | DOI | MR | Zbl

[15] N.V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Math., 96, AMS, Providence, Rhode Island, 2008 | DOI | MR | Zbl

[16] E.M. Landis, Second order equations of elliptic and parabolic types, Nauka, M., 1971 (in Russian) | MR

[17] V.P. Mikhailov, Partial differential equations, Nauka, M., 1976 (in Russian) | MR

[18] V.P. Mikhailov, “The Dirichlet problem for a parabolic equation. I”, Mat. Sb. (N.S.), 61(103):1 (1963), 40–64 (in Russian) | MR

[19] J. Hadamard, Lectures on Cauchy`s problem in linear partial differential equations, Yale Univ. Press, New Haven–London, 1923 | MR | Zbl

[20] A.G. Sveshnikov, A.N. Bogolyubov, V.V. Kravtsov, Lectures on mathematical physics, Nauka, M., 2004 | MR

[21] N.N. Tarkhanov, Complexes of differential operators, Kluwer Ac. Publ., Dordrecht, 1995 | MR | Zbl

[22] G.M. Goluzin, V.I. Krylov, “Verallgemeinerung einer Formel von Carleman und ihre Anwendung auf analytische Fortsetzung”, Mat. Sb., 40:2 (1933), 144–149 (in Russian) | Zbl

[23] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983 | MR | Zbl

[24] A. Aronszajn, C. Treese, L. Lipkin, Polyharmonic functions, Clarendon Press, Oxford, 1983 | MR | Zbl

[25] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéare, Dunod/Gauthier-Villars, Paris, 1969 | MR

[26] S.D. Eidel'man, “Parabolic equations”, Partial differential equations-6, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 63, VINITI, M., 1990, 201–313 (in Russian)