The L^p Neumann problem for the Stokes system in nonsmooth domains

2026/07/16 by Qianyun Miao, Fa Peng
#math.AP

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Abstract

We study the L^p Neumann problem for the Stokes system on bounded Lipschitz domains in \mathbb Rⁿ with n≥ 2. Our main contribution is to introduce a nonlinear gradient quantity -- namely, the linear gradient weighted by a suitable power of the pair (∇ 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≥ 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 W²L^n-1,∞ for n≥ 3, or W²L^1,∞log L for n=2. In particular, our results cover all W^2,q domains with q>n-1.

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