Generalized resolvent estimates for the Stokes system in bounded and unbounded domains
Reinhard Farwig, Hermann Sohr · Journal of the Mathematical Society of Japan · 1994
R. FARWIG and H. SOHR -Solving (1.1) yields an important special class of solutions $u\in W^{2q}(\Omega)^{n}$ $\cap W_{0}^{1.q}(\Omega)^{n}$ of the divergence problem $divu=g$ .TO describe our main results we formulate some assumptions on the domains under consideration and introduce some notations.In this paper, $\Omega$ is the $R^{n}$ , the half space $R_{+}^{n}=\{x=(x', x_{n})\in R^{n} ; x_{n}>0\}$ , a bounded or exterior domain or a domain which is obtained from the half space by a perturbation within a finite region.In Section 3 we also consider the bended half space $H_{\omega}$ defined by $x_{n}>\omega(x_{1}, \cdots x_{n-1})$ where $||\partial_{i}\omega||_{\infty},$ $i=1,$ $\cdots$ , $n-1$ , are sufficiently small.The precise description for $\Omega eq R^{n}$ and $\Omega eq H_{\omega}$ reads as follows.ASSUMPTION 1.1.Let $\Omega\subseteqq R^{n},$ $n\geqq 2$ , be a domain with boundary $\partial\Omega\in C^{1,1}$ and suppose one of the following cases:The last condition (iii) means that $\Omega$ behaves like $R_{+}^{n}$ for sufficiently large $|x|$ .The assumption $\partial\Omega\in C^{1.1}$ means that for each $x\in\partial\Omega$ there exists an open ball $B_{x}$ centered at $x$ and a function $\omega\in C^{1,1}(G)$ on some domain $G\subseteqq R^{n-1}$ such that after a rotation of the Cartesian coordinates, if necessary, the following holds: $y_{n}>\omega(y')$ for all $(y', y_{n})\in\Omega\cap B_{x},$ $y_{n}<\omega(y')$ for all $(y', y_{n})\in(R^{n}\backslash \overline{\Omega})\cap B_{x}$ and $y_{n}=\omega(y')$ for all $(y', y_{n})\in(\partial\Omega)\cap B_{x}$ , where $y'=(y_{1}, \cdots , y_{n-1})$ .We will use the standard notations $L^{q}(\Omega)$ with norm $||\cdot||_{L^{q}(\Omega)}$ (or $||\cdot||_{q}$ if the underlying domain is known from the context), $L_{1OC}^{q}(\Omega),$ $L_{\iota oc}^{q}(\overline{\Omega})$ and $W^{1q}(\Omega)$ , $W_{0}^{1.q}(\Omega),$$W^{2,q}(\Omega)$ , etc. for Sobolev spaces of scalar functions.In particular, $u\in$ $L_{1}^{q_{OC}}(\overline{\Omega})$ where 9 is the closure of $\Omega$ means that $u\in L^{q}(\Omega\cap B)$ for all balls $B$ with $\Omega\cap B eq\emptyset$ .For vector-valued functions in $L^{q}(\Omega)^{n}$ , etc. we will use the same symbol $||\cdot||_{q}$ for the $L^{q}$ -norm; more generally $||(f_{1}, \cdots , f_{m})||_{q}=(\Sigma_{\ell=1}^{m}||f_{i}||_{q}^{q})^{1/Q}$ for $f_{i}\in L^{q}(\Omega)$ or $L^{q}(\Omega)^{n}$ , etc..For $1<q<\infty$ let $q'$ denote the dual exponent, $i$ .$e.,$ $1/q+1/q'=1$ , and let $\langle\cdot, \cdot\rangle$ denote the $L^{q}-L^{q'}$ -pairing of scalar, vector or matrix functions on $\Omega$ .If $X$ is a Banach space and $x*$ its dual space, then we write $[x^{*}, x]$ for the evaluation of $x^{*}\in X^{*}$ in $x\in X$ .However for the trace space $W^{1-1/qq}(\partial\Omega)$ and its dual $W^{-1/q',q'}(\partial\Omega)$ we use $[\cdot, ]_{\partial\Omega}$ .Further let $\partial_{i}=$ $\partial/\partial x_{i},$ $i=1,$ $\cdots$ , $n,$ $ abla=(\partial_{1}, \cdots , \partial_{n}),$ $\Delta=\partial_{1}^{2}+\cdots+\partial_{n}^{2}$ and $ abla^{2}=(\partial_{i}\partial_{j})_{i.j=1}^{n}$ .Finally let $9(\Delta_{q})=W^{2q}(\Omega)\cap W_{0}^{1q}(\Omega)$