Closedness of sums of unbounded operators acting on different variables in the spaces of square-integrable functions of several variables

D. Yu. Ivanov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 45 (2017), pp. 35-48

Résumé

Let $\Omega'$ and $\Omega''$ be some Lebesgue measurable sets in the metric spaces $\mathbf{R}^m$ and $\mathbf{R}^n$, respectively, and $\mathbf{A}$ be a linear operator in the space $L_2(\Omega')$. We define an operator $\tilde{\mathbf{A}}$ in the space $L_2(\Omega'\times \Omega'')$ on the basis of equalities $$ (\tilde{\mathbf{A}}\mathbf{f})(x'')=\mathbf{A}\mathbf{f}(x'')\quad (\mathbf{f}\in D(\tilde{\mathbf{A}}),\ x''\in\Omega''), $$ where $D(\tilde{\mathbf{A}})$ is a domain of operator $\tilde{\mathbf{A}}$. These equations mean that an element $\mathbf{f}\in L_2(\Omega'\times \Omega'')$ represented by a function $\mathbf{f}(x'')$ with values in $D(\mathbf{A})$ belongs to the set $D(\tilde{\mathbf{A}})$ if there exists an element $\mathbf{g}\in L_2(\Omega'\times \Omega'')$ represented by the function $\mathbf{g}(x'')$ such that the pointwise equalities $\mathbf{g}(x'')=\mathbf{A}\mathbf{f}(x'')$ are satisfied almost everywhere in the Lebesgue measure on the set $\Omega''$. Then, $\tilde{\mathbf{A}}\mathbf{f}=\mathbf{g}$. Similarly, using a linear operator $\mathbf{B}$ acting in the space $L_2(\Omega'')$, we define an operator $\tilde{\mathbf{B}}$ in the space $L_2(\Omega'\times \Omega'')$. It is proved that the sum of operators $\tilde{\mathbf{A}}+\tilde{\mathbf{B}}$ defined on the set $D(\tilde{\mathbf{A}})\cap D(\tilde{\mathbf{B}})$ is closed if the operators $\mathbf{A}$ and $\mathbf{B}$ are generators of some $C_0$-semigroups of contractions; here, the operator $\mathbf{B}$ is selfadjoint and has a purely point spectrum. For example, if the operator $\mathbf{A}_t$, $(\mathbf{A}_t\mathbf{f})(t)=f'(t)$ is defined on absolutely continuous functions $f(t)\in L_2(I_T)$ ($I_T\equiv [0, T]$) such that $f'(t)\in L_2(I_T)$ and $f(0)=0$, as well as on equivalent functions and operator $\mathbf{B}_y$, $(\mathbf{B}_y\mathbf{f})(y)=-f''(y)$, is defined on absolutely continuously differentiable functions $f(y)\in L_2(I_Y)$ ($I_Y\equiv [0, Y]$) such that $f''(y)\in L_2$ and $f'(0)-\lambda_0 f(0)=0$, $f'(0)+\lambda_Yf(0)=0$ ($0\leqslant \lambda_0$, $\lambda_Y\leqslant\infty$), as well as on equivalent functions, the sum of differential operators $\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y$ is closed. The closure of the operator $\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y$ is used as a coefficient in operator-differential equations in the formulation of problems of multidimensional non-stationary heat conduction. We have studied smoothness of functions included in the domains of powers of operators $\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y$. It is proved that if $f(y, t)\in D\left((\tilde{\mathbf{A}}_t+\tilde{\mathbf{B}}_y)^n\right)$ ($n\geqslant 2$), then, almost everywhere on the set $I_Y\times I_T$, there exist derivatives $\partial_t^{l-1}\partial_y^{2(n-l)-1}f$ ($l=\overline{1, n-1}$) equivalent to functions absolutely continuous on $I_Y\times I_T$.

Détail

Keywords: closed linear operator, sum of operators, generator of $C_0$-semigroup, domain of definition of operator.

Bibliographie

[1] Reed M., Simon B., Methods of Modern Mathematical Physics, v. 2, Fourier Analysis, Self-adjointness, Academic Press, New York–London, 1975 | MR | MR | Zbl

[2] Clement Ph., Heijmans H. J. A. M., Angenent S., Duijn C. J., van and Pagter B. de, One-parameter Semigroups, CWI Monographs, 5, North Holland, Amsterdam–New York, 1987 | MR | MR | Zbl

[3] Ivanov D. Yu., “Solution of two-dimensional boundary-value problems corresponding to initial-boundary value problems of diffusion on a right cylinder”, Differential Equations, 46:8 (2010), 1104–1113 | DOI | MR | Zbl

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

[5] Ivanov D. Yu., “An economical method of the calculation of operators resolving some heat conduction problems in straight cylinders”, Actual Problems of Humanitarian and Natural Sciences, 2014, no. 9, 16–32

[6] Ivanov D. Yu., “Calculation of operators resolving problems of heat conduction in straight cylinders using the semigroup symmetry”, Proceedings of Moscow State Technical University MAMI, 4:4(22) (2014), 26–38

[7] Ivanov D. Yu., “Stable solvability in spaces of differentiable functions of some two-dimensional integral equations of heat conduction with an operator-semigroup kernel”, Tomsk State University Journal of Mathematics and Mechanics, 2015, no. 6(38), 33–45 | DOI

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

[9] Lyantse V. E., Storozh O. G., Methods of the Theory of Unbounded Operators, Naukova Dumka, Kiev, 1983

[10] Reed M., Simon B., Methods of Modern Mathematical Physics, v. 4, Analysis of Operators, Academic Press, New York–London, 1978 | MR | MR | Zbl

[11] Kreyn S. G., Linear Differential Equations in the Banach Space, Nauka, M., 1967, 464 pp.

[12] Kato T., Perturbation theory for linear operators, Springer Verlag, Berlin–New York, 1966 | MR | MR | Zbl

[13] Budak B. M., Samarskiy A. A., Tikhonov A. N., Collection of Problems in Mathematical Physics, Nauka, M., 1980, 685 pp. | MR

[14] Vilenkin N. Ya., Gorin E. A., Kostyuchenko A. G., Krasnosel'skiy M. A., Kreyn S. G. et al., Functional Analysis, Reference mathematical library, ed. Kreyn S. G., Nauka, M., 1964, 425 pp.

[15] Smirnov V. I., A Course of Higher Mathematics, v. 5, St. Publ. Phis. Math. Lit., M., 1959, 655 pp.

[16] Kolmogorov A. N., Fomin S. V., Elements of the Theory of Functions and Functional Analysis, Nauka, M., 1976, 543 pp. | MR