Global second order optimal regularity for the vectorial $p$-Laplacian

TL;DR

This work advances the global second-order regularity theory for vectorial $p$-Laplacian systems by proving sharp stress-field estimates up to the boundary under reduced domain regularity, captured through Lorentz-Zygmund boundary conditions. The authors develop a robust regularization and domain-exhaustion framework to derive $|D\boldsymbol u|^{\gamma-1}D\boldsymbol u\in W^{1,2}(\Omega)$ for $\gamma\ge(p-\alpha)/2$ with $0\le \alpha<h(p)$ (or $\alpha<1$ for larger $p$), which yields $\boldsymbol u\in W^{2,2}(\Omega)$ when $1<p<3$ and extends to convex domains with no extra boundary assumptions. A key novelty is the combination of boundary curvature control via $\partial\Omega\in W^2X$, Lorentz-type boundary spaces, and a nonlinear Calderón–Zygmund type analysis adapted to the vectorial setting. The paper also derives global integrability of $1/|D\boldsymbol u|^{\sigma}$ under a sign condition on $\mathbf f$, leading to improved Sobolev regularity $\boldsymbol u\in W^{2,q}(\Omega)$ for $p\ge 3$. These results significantly refine the understanding of boundary regularity for vector-valued $p$-Laplacian problems and extend sharp global estimates beyond smooth domains.

Abstract

We obtain optimal regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol Δ}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x)\,\, \mbox{ in $Ω$}\,.$$ More precisely we address the issue of global second order estimates for the stress field.

Theorems & Definitions (27)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Theorem 1.5: Convex domains
• Theorem 1.6
• Remark 1.7
• Theorem 1.8: Convex domains
• Theorem 1.9: Local case
• Remark 1.10