Some new types of strengthened Sobolev spaces and their applications to the theory of nonstationary problems. (Q1432253)

Let \(\Omega \) be a bounded domain in \(\mathbb{R}^2\) with a Lipschitzian piecewise smooth boundary \(\Gamma \) and let \(Q_t\) be the cylinder \(\bar{\Omega} \times [0,t].\) Let \(S_t \subset \bar{Q_t}\) consists of a finite number of vertical rectangles \(F_r= S_r \times [O,t]\) where \(S_r\) is a segment in \(\bar{\Omega}.\) The author introduces different energy Hilbert spaces \(\mathcal{E}\) with special traces on the rectangles \(F_r.\) Several properties of these spaces are analyzed, in particular the normal invertibility of the trace operator on \(S_t\) and the density of smooth functions in \(\mathcal{E}.\) Applications to time-dependent problems are considered. Previous results of the author on different types of strengthened Sobolev spaces are used throughout this note.