On the projecting in the space of solenoidal vector fields

M. I. Belishev   ; A. K. Glasman

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 28, Tome 257 (1999), pp. 16-43

Résumé

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.