I was reading this paper ([arXiv link](https://arxiv.org/abs/0912.3506))



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> On the Large Time Behavior of Solutions of the Dirichlet problem for Subquadratic Viscous Hamilton-Jacobi Equations
> Guy Barles (LMPT), Alessio Porretta, Thierry Wilfried Tabet Tchamba (LMPT)
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The authors claimed that in general if $f\in W^{1,\infty}(B(0,2)),1<m\leq 2$ and $u\in W^{2,p}\_{loc}(B(0,2))$ is a solution to
$$ |Du|^m - f(x) - \Delta u = 0 \qquad{in}\;B(0,2)$$
then



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> If $|Du|\_{L^\infty(B(0,1/2))}\leq K$ then $|D^2u|\_{L^\infty(B(0,1/2))}\leq K'$ by standard elliptic theory.
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My questions are:


* What is the exact argument did they use to arrive at this conclusion?
* Would the argument hold true if I have
$$ \delta u+ |Du|^m - f(x) - \Delta u = 0 \qquad{in}\;B(0,2)$$
for $\delta > 0$ instead?



A regularity estimate for second-derivative
The main tools are the following elliptic regularity results: assume that $u \in W^{2,p}\_{loc}$ and let $f=\Delta u$.


a) If $f \in L^q\_{loc}$ with $q>p$, then $u \in W^{2,q}\_{loc}$;


b) If $f \in W^{1,p}\_{loc}$, then $u \in W^{3,p}\_{loc}$.


With this in mind, first note that $|Du|^m \in W^{1,p}\_{loc}$, because $m>1$ and $Du$ is bounded. This gives $u \in W^{3,p}\_{loc}$ and the result follows if $p>N$, by Sobolev embedding. If, instead, $p \leq N$, use first Sobolev embedding to get $u \in W^{2,q}\_{loc}$ with any $q<\infty$ if $p=N$ and $1/q=1/p-1/N$ when $p<N$ and then rerun the argument before with $q$ instead of $p$. In a finite number of steps you get the result.


