Today, we prove a very interesting inequality along the boundary of a convex set. This proof is adapted from a book by P.L. Lions entitled Generalized Solutions of Hamilton-Jacobi Equations in the series Research Notes In Mathematics Series by Pitman Publishing.
Theorem. Assume that is convex domain and . is the outward unit normal vector field along the boundary . We assume further that the function satisfies
Then we have the following inequality
Proof. The proof is surprisingly simple. First, we can assume that is a sublevel of some convex function, say , such that and along , i.e.
Then the outward unit normal vector field along the boundary can be calculated using
Note that this is clear since the boundary can be consider as the level set of at the level . Therefore, the gradient is automatically perpendicular to the level set (and in this case, just the boundary ). (See a similar argument I claimed in this topic about the Pohozaev equality.)
Since , we know that . Taking derivatives with respect to then multiplying by , and finally summing from to , we obtain
In fact, Lions claimed that is parallel to the normal , that is to say
for some . In particular,
But I am not sure how to prove it.
Hence, we deduce that
since is convex. The reason is that is convex if and only if its Hessian is positive definite, i.e.
Apparently, this result looks like the Hopf lemma: If is positive in the interior of and vanishes on , then achieves its minimum at any point on the boundary , consequently, since is falling downward when approaching the boundary. In our new context, the function is obviously positive in and on the boundary the condition is “similar to” , then one expect that along the boundary. Since the precise equality is far from the truth, one needs to assume that the domain is convex and I guess this is a way around to guarantee in some sense.
For certain , we could have something on as well.