平面Hölder- $α$域的散度的右逆
A right inverse of the divergence for planar Hölder-$α$ domains
摘要：
当$Ω\subset\R^n$是有界区域时，$f\in L^2(Ω)$的${div} {\bf u} = f$的解${\bf u}\in H^1_0(Ω)^n$的存在性是Stokes方程分析中的一个基本结果.特别是它允许显示存在的解决方案$({\bf u},p)\in H^1_0(Ω)^n\times L^2(Ω)$，其中${\bf u}$是速度和$p$的压力。已知当$Ω$是Lipschitz整环时，上述结果成立，而对任意Hölder-$α$整环不成立.本文证明了：若$Ω$是平面单连通Hölder-$α$整环，则在适当的权空间中存在连续的散度右逆，其中权是到边界距离的幂.此外，我们还证明了所得结果中距离的幂是最优的。在我们的结果中，零边界条件被替换为一个较弱的。对于具有幂型外尖点的特殊情形，我们证明了我们的弱边界条件与标准边界条件是等价的。在这种情况下，我们证明了适当的加权Sobolev空间中的斯托克斯方程的适定性，从而获得了某些$r<2$的解$({\bf u},p)\in H^1_0(Ω)^n\times L^r(Ω)$的存在性取决于尖点的功率。
英文摘要：
If $Ω\subset\R^n$ is a bounded domain, the existence of solutions ${\bf u}\in H^1_0(Ω)^n$ of ${div} {\bf u} = f$ for $f\in L^2(Ω)$ with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution $({\bf u},p)\in H^1_0(Ω)^n\times L^2(Ω)$, where ${\bf u}$ is the velocity and $p$ the pressure. It is known that the above mentioned result holds when $Ω$ is a Lipschitz domain and that it is not valid for arbitrary Hölder-$α$ domains. In this paper we prove that if $Ω$ is a planar simply connected Hölder-$α$ domain, there exist right inverses of the divergence which are continuous in appropriate weighted spaces, where the weights are powers of the distance to the boundary. Moreover, we show that the powers of the distance in the results obtained are optimal. In our results, the zero boundary condition is replaced by a weaker one. For the particular case of domains with an external cusp of power type, we prove that our weaker boundary condition is equivalent to the standard one. In this case we show the well posedness of the Stokes equations in appropriate weighted Sobolev spaces obtaining as a consequence the existence of a solution $({\bf u},p)\in H^1_0(Ω)^n\times L^r(Ω)$ for some $r<2$ depending on the power of the cusp.
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