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 $q>n-1$.
In this paper, we develop the method of moving planes entirely in the weak formulation for the fractional $p$-Laplacian in the singular range $1<p\leq2$. We first establish a small region principle for antisymmetric functions and apply it to prove radial symmetry and monotonicity of nonnegative weak solutions of fracti...
We consider positive weak solutions of $-\Delta_p u=f(u)$, with $f$ positive and locally Lipschitz continuous. In the singular case $1<p<2$, the Harnack-type inequalities, the strong maximum principle for the linearized operator, the description of the critical set and the strong comparison principle established by Dam...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
By refining the approach developed in \cite{AGH-2015, GS2026a}, we establish resolvent estimates in $L^p_\sigma (\Omega)$ for the Stokes operator in a bounded Lipschitz domain $\Omega$ in $\R^3$ for any $(3/2)-\e<p\le \infty$, where $\e>0$ depends on $\Omega$. As a consequence, the Stokes operator generates a uniformly...
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their H\"{o}lder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcom...
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
Fan-Ze Kong, Xiao-Yu Zeng· 0 citations
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