# Viscosity solution of the PDE

Let $\Omega$ be bounded domain, $u=0$ on $\delta\Omega$ and $$|Du|-f(x,u)=0$$

where $f\ge 0$ and $f$ is strictly monotone for fixed $x.$ I am looking for the reference to show that it has unique viscosity solution.

P.S. It has been posted before mathstackexchange.

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What is $|Du|$? – Andrey Rekalo Jan 14 '13 at 14:49
This stands for gradient of $u.$ – nick Jan 14 '13 at 14:57
This is literally the first example I saw when I studied viscosity solutions in grad school. Have you checked the users guide to viscosity solutions for a general existence and unique theorem? – Luis Silvestre Jan 15 '13 at 17:27
The answer already appeared here: math.stackexchange.com/questions/278384/viscosity-solutions/… – nick Jan 15 '13 at 23:41
Yes, but the proof in math.stackexchange is wrong because the functions $\phi$ may not exist if $u_1$ and $u_2$ are not differentiable at $x_0$. The classical proof uses the doubling of variables method, very much like in the proof of Theorem 1 in section 10.2 of Evans book. – Luis Silvestre Jan 16 '13 at 0:45