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.