AI 中文总结
研究\(\mathbb{R}^n\)中有界利普希茨区域上斯托克斯系统的\(L^p\)诺伊曼问题，引入新的非线性梯度量建立全局二阶估计，改进反向赫尔德不等式，扩展可解性范围，方法还适用于半凸及更一般利普希茨区域。
AI 中文摘要
我们研究\(\mathbb{R}^n\)（\(n\geq2\)）中有界利普希茨区域上斯托克斯系统的\(L^p\)诺伊曼问题。主要贡献是引入非线性梯度量，即用\((\nabla u,\phi)\)的合适幂次加权的线性梯度取代耿和沈（2025）先前工作中使用的标准线性梯度。此新方法使我们能建立全局二阶估计，进而得到改进的反向赫尔德不等式，扩展了凸区域已知的可解性范围，尤其改进了\(n\geq3\)时的上界。在非凸情形下，该方法也适用于半凸区域。对于更一般的利普希茨区域，在边界第二基本形式的小性条件下证明了可解性，假设边界在\(n\geq3\)时在弱型洛伦兹空间\(W^2L^{n - 1,\infty}\)中具有二阶导数，或在\(n = 2\)时在\(W^2L^{1,\infty}\log L\)中。特别地，结果涵盖所有\(q>n - 1\)的\(W^{2,q}\)区域。
英文摘要
We study the $L^p$ Neumann problem for the Stokes system on bounded Lipschitz domains in $\mathbb R^n$ with $n\ge 2$. Our main contribution is to introduce a nonlinear gradient quantity -- namely, the linear gradient weighted by a suitable power of the pair $(\nabla u,ϕ)$ -- in place of the standard linear gradient used in previous work by Geng and Shen (2025). This new approach allows us to establish a global second-order estimate, which in turn yields an improved reverse Hölder inequality and extends the known range of solvability for convex domains, particularly improving the upper bound for $n\ge 3$. Beyond the convex setting, our method also applies to semi-convex domains. Moreover, for more general Lipschitz domains, we prove solvability under a smallness condition on the second fundamental form of the boundary, assuming the boundary has second-order derivatives in the weak-type Lorentz spaces $W^2L^{n-1,\infty}$ for $n\ge 3$, or $W^2L^{1,\infty}\log L$ for $n=2$. In particular, our results cover all $W^{2,q}$ domains with $q>n-1$.
