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Zapiski Nauchnykh Seminarov POMI, 1999, Volume 257, Pages 16–43
(Mi znsl981)
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This article is cited in 3 scientific papers (total in 3 papers)
On the projecting in the space of solenoidal vector fields
M. I. Belisheva, A. K. Glasmanb a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
b Saint-Petersburg State University
Abstract:
Let $\Omega\subset\mathbf R^3$ be a bounded domain; let $\Omega^\xi:=\{x\in\Omega\mid\operatorname{dist}(x,\partial\Omega)<\xi\},\xi>0$ be an increasing family of subdomains; let $\varepsilon=\varepsilon(x)$ be a positive function in $\overline{\Omega}$; $\mathscr H:=\{\bold y=\bold y(x)\mid\int_\Omega dx\varepsilon|\bold y|^2<\infty,\,\mathrm {div}\,\varepsilon\bold y=0$ in ${\Omega}\}$ be a space of $\varepsilon$-solenoidal vector fields; let $\mathscr H^\xi:=\{\bold y\in\mathscr H\mid\mathrm {supp}\,\bold y\subset\overline{\Omega^\xi}\}$, $\xi>0$ be a family of subspaces; let $G^{\xi}$ be orthogonal projectors in $\mathscr H$ onto $\mathscr H^\xi$. The unitary transform which diagonalizes the family of projectors $\{G^\xi\}$ is constructed: it transfers $\int\xi dG^\xi$ into an operator multiplying by independent variable. An isometry of the transform is proved with the help of the operator Riccati equation for the Neumann–to–Dirichlet map.
Received: 20.11.1998
Citation:
M. I. Belishev, A. K. Glasman, “On the projecting in the space of solenoidal vector fields”, Mathematical problems in the theory of wave propagation. Part 28, Zap. Nauchn. Sem. POMI, 257, POMI, St. Petersburg, 1999, 16–43; J. Math. Sci. (New York), 108:5 (2002), 642–664
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https://www.mathnet.ru/eng/znsl981 https://www.mathnet.ru/eng/znsl/v257/p16
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