Third order estimates and the regularity of the stress field for solutions to $p$-Laplace equations

We consider solutions to $$-{\mathrm{\Delta }}_{p}u=f(x)\text{in }\mathrm{\Omega },$$ when $p$ approaches the semilinear limiting case $p=2$ and we get third order estimates. As a consequence we deduce improved regularity properties of the stress field.