For quasilinear equations -div(a(|∇u|)∇u)=f, the stress field a(|∇u|)^k∇u is shown to belong to W^{1,2}(Ω) up to the boundary for the optimal range of k, under weak W^{2,X} boundary regularity, with a convex-domain version requiring no boundary regularity.
Global second order optimal regularity for the vectorial $p$-Laplacian
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abstract
We obtain optimal regularity results for solutions to vectorial $p$-Laplace equations $$ -{\boldsymbol \Delta}_p{\boldsymbol u}=-\operatorname{\bf div}(|D{\boldsymbol u}|^{p-2}D{\boldsymbol u}) = {\boldsymbol f}(x)\,\, \mbox{ in $\Omega$}\,.$$ More precisely we address the issue of global second order estimates for the stress field.
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Second-order boundary estimates for solutions to a class of quasilinear elliptic equations
For quasilinear equations -div(a(|∇u|)∇u)=f, the stress field a(|∇u|)^k∇u is shown to belong to W^{1,2}(Ω) up to the boundary for the optimal range of k, under weak W^{2,X} boundary regularity, with a convex-domain version requiring no boundary regularity.