The $L^p$ Neumann problem for the Stokes system in nonsmooth domains
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 $(\nabla u,\phi)$ -- 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 H\"older 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 $q>n-1$.