Solution of boundary problems for a two-dimensional elliptic operatordifferential equation in an abstract Hilbert space using the method of boundary integral equations

Ivanov D.Yu.

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 60 (2019), pp. 11-31

Résumé

In this paper, we study boundary-value problems of the first, second, and third kinds for the differential-operator equation $\Delta_2\mathbf{u} = \mathbf{Bu}$ ($\Delta_2\equiv\partial_{x_1x_1}^2+\partial_{x_2x_2}^2$) in an open two-dimensional bounded simply connected domain $\Omega^+$ or its open exterior $\Omega^-$. Here, $\mathbf{u}(x_1,x_2)$ is a vector function with values in an abstract Hilbert space $H$; $\mathbf{B}$ is a linear closed densely operator defined in the space $H$ and generating an exponentially decreasing $C_0$-semigroup of contractions $\mathbf{T}(\tau)$: $||\mathbf{T}(\tau)||\leqslant \exp(-p\tau)$ ($p > 0$). Solutions of the boundary-value problems are obtained in the form of vector potentials with unknown vector functions similar to density functions, which are found from Fredholm boundary integral equations of the second kind, wherein kernels of integral operators are expressed through the $C_0$-semigroup $\mathbf{T}(\tau)$. Let $\partial\Omega$ be the boundary of the domain $\Omega^{\pm}$. Under the condition $\partial\Omega\in C^2$, the stable solvability of the boundary-value problems in the space $C(\overline{\Omega^{\pm}};H)$ is proved. Here, $C(\overline{\Omega^{\pm}};H)$ is the Banach space of vector functions, continuous on the closed set $\overline{\Omega^{\pm}}$ with values in the space $H$. The stable solvability of the boundary integral equations in the spaces $L_2(\partial\Omega; H)$ and $C^k(\partial\Omega; H^n_{\mathbf{B}})$ ($k, n\geqslant0$) is also proved under the conditions $\partial\Omega\in C^2$ and $\partial\Omega\in C^{k+2}$, respectively. Here, $L_2(\partial\Omega; H)$ is the Hilbert space of vector functions, square-summable on the set $\partial\Omega$ with values in the space $H$; $C^k(\partial\Omega; H^n_{\mathbf{B}})$ is the Banach space of vector functions, $k$ times continuously differentiable on the set $\partial\Omega$ with values in the Sobolev type space $H^n_{\mathbf{B}}$ defined by powers $n+1$ of the operator $\mathbf{B}$.

Détail

Keywords: Boundary-value problem, differential-operator equation, boundary integral equation, semigroup of operators, generator, vector-valued function, operator-valued function, unitary dilation.

Bibliographie

[1] G. N. Polozhii, Equations of mathematical physics, Vysshaya Shkola, M., 1964, 560 pp. | MR

[2] S. G. Krein, Linear differential equations in Banach space, American Math. Soc., Providence, 1971, 390 pp. | MR

[3] V. K. Ivanov, I. V. Mel'nikova, A. I. Filinkov, Operator-differential equations and ill-posed problems, Nauka, M., 1995, 176 pp.

[4] S. Yakubov, Ya. Yakubov, Differential-operator equations. Ordinary and partial differential equations, Chapman and Hall/CRC, Boca Raton, 2000, 541 pp. | MR | Zbl

[5] V. B. Shakhmurov, “Maximal regular abstract elliptic equations and applications”, Siberian Mathematical Journal, 51:5 (2010), 935–948 | DOI | MR | Zbl

[6] V. Shakhmurov, Singular degenerate problems and application, arXiv: (submitted on 5 July 2017) 1707.01376

[7] V. Shakhmurov, Regularity properties of singular degenerate abstract differential equations and applications [Elektronnyi resurs], arXiv: (submitted on 5 July 2017) 1707.01771 | MR

[8] D. Y. Ivanov, Differential Equations, 46:8 (2010), 1104–1113 | DOI | MR | Zbl

[9] D. Yu. Ivanov, R. I. Dzerzhinskiy, “Solution of the Robin problems for two-dimensional differential-operator equations describing the thermal conductivity in a straight cylinder”, Scientific and Technical Journal of the Volga Region, 2016, no. 1, 15–17

[10] D. Yu. Ivanov, “Using of vector potentials for solving the two-dimensional Robin problem describing the heat conductivity in a straight cylinder”, Actual Problems of Humanitarian and Natural Sciences, 2016, no. 3-1, 8–14

[11] D. Yu. Ivanov, “Stable solvability in spaces of differentiable functions of some twodimensional integral equations of heat conduction with an operator-semigroup kernel”, Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 6, 33–45 | DOI

[12] Sz. Nagy B., C. Foias, Harmonic analysis of operators on Hilbert space, Akademiai Kiado, Budapest, Amsterdam–London, 1970, 389 pp. | MR | MR | Zbl

[13] S. G. Mikhlin, Mathematical physics: an advanced course, North-Holland Pub. Co., Amserdam, 1970, 561 pp. | MR | MR | Zbl

[14] N. Dunford, J. T. Schwartz, Linear operators, v. II, Spectral theory, Self adjoint operators in Hilbert space, Interscience Publishers, New York–London, 1963, 1065 pp. | MR | Zbl

[15] N. Dunford, J. T. Schwartz, Linear operators, v. I, General theory, Interscience Publishers, New York–London, 1958, 858 pp. | MR | Zbl

[16] T. Kato, Perturbation theory for linear operators, Springer, Berlin–Heidelberg–New York, 1976, 620 pp. | MR | Zbl

[17] M. Reed, B. Simon, Methods of Modern Mathematical Physics, v. 1, Functional Analysis, Academic Press, New York–London, 1972, 325 pp. | MR | Zbl

[18] V. I. Smirnov, A course of higher mathematics, v. 4, Integral equations and partial differential equations, Pergamon Press, Oxford–New-York, 1964, 811 pp. | MR | MR

[19] D. Yu. Ivanov, “Validation of a numerical algorithm for solving inverse boundary value problems for heat equation that takes into account the semigroup symmetry”, Computational Mathematics and Mathematical Physics, 38:12 (1998), 1948–1961 | MR | Zbl