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This may be an embarrassing question but could any one please tell me if we have a comparison principle for the viscosity solution of the following equation

$$ \begin{cases} -\nabla\cdot A(x)\nabla u(x) + b(x)\cdot \nabla u(x) = f(x), \quad x\in \Omega \\ u(x)=g(x), \quad x\in \partial \Omega \end{cases}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^n$?

In the user's guide of viscosity solutions, though the authors treat fully nonlinear equations, it's not clear to me that this simple linear PDE would have a comparison principle since the assumption for $$F(x,u,Du,D^2u)$$ is so that there exists a $\gamma>0$ so that $$F(x,u,p,X) - F(x,v,p,X) \ge \gamma(u-v)$$ if $u>v$.

Please enlighten my ignorance!

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Definitely not an embarrassing question. The answer is that comparison may not follow from the user's guide, but the condition you state is not the issue. The more problematic assumption is (3.14) in the user's guide, which requires regularity of the PDE itself. This only holds when $A$ is constant (and positive definite), and $b \in C^1$.

If the PDE is uniformly elliptic ($A \geq cI$, $c>0$), then you do not need the condition you stated (3.13 in User's guide) for comparison to hold. The condition (3.13) was used for convenience in the user's guide, and is used to perturb a subsolution to a strict subsolution, since comparison is easier to prove when one of the sub/super solutions is strict. This is embedded in the proof of the comparison principle (in the doubling variables part) and leads to the contradiction.

When the PDE is uniformly elliptic (and linear), you can do this strictification step by adding a small and very convex function (like $\epsilon e^{\beta x_1}$) to the subsolution (as in Evans Chapter 6, classical maximum principle). There are other cases that can be handled with other tricks, such as the eikonal equation $\|\nabla u\|=f$. There is some discussion of this in section 5.C in the user's guide, but they do not mention the tricks for the strictification step.

All this being said, if the equation is linear and uniformly elliptic, and $A,b$ are sufficiently regular, then the viscosity solutions are classical, by elliptic regularity, so the classical comparison principle holds. The theory of viscosity solutions was not designed for linear elliptic equations (since these were already "solved"), hence they are not dwelled on in the user's guide.

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  • $\begingroup$ Thank you, Jeff! A friend of mine showed me the trick you mentioned that was in Evans and that was very nice. $\endgroup$
    – S.V.
    Commented Feb 2, 2018 at 5:29

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