MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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:… – 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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.