Weighted Lq-theory for the Stokes resolvent in exterior domains
Reinhard Farwig, Hermann Sohr · Journal of the Mathematical Society of Japan · 1997
In an exterior domain $\Omega\subset R^{n},$ $n\geqq 2$ , consider the generalized Stokes resolvent system $\lambda u-\Delta u+ abla p=f$ R. FARWIG and H. SOHR [7,14,17,25] for results in spaces without weights.For the special weights of the form $(1+|x|)^{\alpha}$ a descriPtion of the Helmholtz decomPosition has been given in [21] even without the restriction on a below.In this Pape $r$ we solve the generalized Stokes resolvent problem(1.1)with $g eq 0$ and construct the Helmholtz decomposition in weighted $L^{q}$ -spaces for a large class of weights.A weight function $0\leqq w\in L_{1oc}^{1}(R^{n})$ is said to be of the Muckenhoupt class $\mathcal{A}_{q}$ iff $\sup_{Q}(\frac{1}{|Q|}\int_{Q}wdx)\cdot(\frac{1}{|Q|}\int_{Q}w^{-1/(q-1)}dx)^{q-1}\leqq C<\infty$ ; here $1<q<\infty$ and $Q\subset R^{n}$ runs tbrough the se $t$ of all bounded cubes of $R^{n}$ with axes parallel to the coordinate axes, and $|Q|$ denotes the Lebesgue measure of $Q$ .Examples are given by the standard radial weights with fixed $x_{0}\in R^{n}$ defined by $w(x)=|x-x_{0}|^{\alpha}$ or $w(x)=(1+|x|)^{a}$ , $-n<a<n(q-1)$ ; also finite positive sums of these terms multiplied by logarithmic terms $\log^{\beta}(2+|x|),$ $\log^{\beta}(2+|x-x_{0}|^{-1})$ , $\beta\in R$ , are allowed.Even the distance dist $(x, M)^{\alpha}$ of $x$ to a bounded manifold $M$ and anisotropic functions such as $(1+|x|)^{\alpha}(1+|x|-x_{1})^{\beta}$ , $x=(x_{1}, \cdots x_{n})\in R^{n}$ , for certain $\alpha,$ $\beta\in R$ define $\mathcal{A}_{q}$ -weights; see Lemma 2.2, 2.3 and Remark 2.4 below for construction and properties of $\mathcal{A}_{q}$ -weights.The reason to consider weights of class $\mathcal{A}_{q}$ is the fact that the classical multiplier theorem of H\"ormander and Michlin remains true in weighted $L^{q}$ -spaces on $R^{n}$ for an $\mathcal{A}_{q}$ -weight.Given a weight $w\in \mathcal{A}_{q}$ define the weighted spaces $L_{w}^{q}( \Omega)=\{u\in L_{1oc}^{1}(\overline{\Omega});||u||_{q,w}=||uw^{1/q}||_{q}=(\int_{\Omega}|u|^{q}wdx)^{11q}<\infty\}$ , $H_{yj^{q}}^{2}(\Omega)=$ { $u\in L_{1oc}^{1}(\overline{\Omega});u$ , Vu, $ abla^{2}u\in L_{w}^{q}(\Omega)$ }for the velocity, and the homogeneous Sobolev space $\dot{H}_{\dot{w}}^{1q}(\Omega)=\{p\in L_{1oc}^{1}(\overline{\Omega}); abla p\in L_{w}^{q}(\Omega)^{n}\}$ ; here $\Omega=R^{n}$ or $\Omega\subset R^{n}$ is an exterior domain.Then the main $re$ sult on the Stokes resolvent problem (with $g=divu=0$) in $R^{n}$ reads as follows:THEOREM 1.1.Let $q\in(1, \infty),$ $n\geqq 2,$ $w\in \mathcal{A}_{q}$ and $\text{\'{e}}\in(O, \pi/2)$ .Then for every $\lambda\in S$ .and $f\in L_{w}^{q}(R^{n})^{n}$ the resolvent problem