In general it is not known that if and is a viscosity solution of and in implies that .

If is uniformly continuous on , then must equal .

Fix any and real number . Choose a function such that

on , on ,

and ,

Denote , otherwise . and . Then if .

Define on as

.

Claim: has a global maximum achieved only if and .

- if , then
- if , then

- if ,

.

If , then

claim holds.

Suppose achieves its maximum at

Then

Define

Then

has a maximum at . Since is the viscosity subsolution of

Similarly we have and has a minimum at

.

Because is uniformly continuous

where is the module of continuity of .

Since going to subsequence if necessary, there exists , such that

as

as .

Letting in (1), we get for some

let , we get .

By symmetry, .

This proof is gleaned from *M.G.Crandall, L.C.Evans and P.L.Lions, Properties of Viscosity solution of Hamilton-Jacobi equations.* And *L.C.Evans, Some man-min methods for the Hamilton-Jacobi equation. *The original proof require to be bounded. Here we remove the requirement.

This problem is a note to *Micheal G. Crandall, Hitoshi Ishii, and Pierre Louis Lions, User’s guide to viscosity solutions of second order partial differential equations.*