Some embedding inequalities for anisotropic Sobolev spaces (Q2755264)

Let \(\Omega\subset \mathbb{R}^{n}\) be a bounded domain and let \(\partial \Omega\) be a boundary of \(\Omega\), \(Q_{T}=\Omega\times(0,T), 0<T<\infty\). Let us suppose that there exists a covering \(\{\Lambda_{j}\}_{j=1}^{M}\) of \(\Omega\) with properties: NEWLINENEWLINENEWLINE1) if \(\partial\Lambda\cap\partial\Omega=\emptyset\) for some \(j\), then \(\Lambda_{j}\) is a \(n\)-dimensional cube with edge \(r_0\); NEWLINENEWLINENEWLINE2) if \(\partial\Lambda\cap\partial\Omega\neq\emptyset\) for some \(j\), then there exists point \(\xi^{(j)}\in \partial\Lambda\cap\partial\Omega\) such that in the local coordinate system with origin in \(\xi^{(j)}\) the set \(\Lambda_{j}\) has the form \(\{y\in R^{n}:|y_{i}|<r_1\), \(i=1,\ldots,n-1\), \(\phi_{j}(y')- 2r_1<y_{n}<\phi_{j}(y')\}\), where \(\phi_{j}\in C^{l_0}(B'_{r_1} (0)\), \(B'_{r_1}(0)=\{y':|y'|<r_1\}\), \(y'=(y_1,\ldots,y_{n-1})\). NEWLINENEWLINENEWLINEThe main result of article is the following.NEWLINENEWLINENEWLINELet \(\partial\Omega\in C^{l_0}\), \(l_0>\max\{bk,1\}\), \(T\leq\min\{r_0^{b},r_1^{b},1\}\), \(q\geq p, u\in W_{p}^{(bk,k),0}(Q_{T})\), \(s,\alpha\) such that \(j=|\alpha|+ bs<bk\). Then there exists constants \(K_1, K_2\) such that: NEWLINENEWLINENEWLINEa) if \(0\leq \rho <bk-j-(1/p-1/q)(n+b)\), then \(({\partial\over\partial t})^{s} D^{\alpha}u\in W_{q}^{(\rho,\rho/b),0}(Q_{T})\) and \(\|({\partial\over\partial t})^{s}D^{\alpha} u\|_{q,Q_{T}}^{(\rho,\rho/b)}\leq K_1 T^{k-(j+\rho)/b-(1/p- 1/q)(n+b)/b}\|u\|_{p,Q_{T}}^{(bk,k)}\); NEWLINENEWLINENEWLINEb) if \(0\leq \rho <bk-j-(n+b)/p\), then \(({\partial\over\partial t})^{s} D^{\alpha}u\in C^{(\rho,\rho/b),0}(Q_{T})\) and NEWLINE\[NEWLINE\left|\left({\partial\over\partial t}\right)^{s}D^{\alpha} u\right|_{Q_{T}}^{(\rho,\rho/b)}\leq K_2 T^{k-(j+\rho)/b- (n+b)/bp}\|u\|_{p,Q_{T}}^{(bk,k)}NEWLINE\]NEWLINE holds true.