Recently, my friend, CR, has shown me an iteration by Stampacchia. Stampacchia proposed his iteration in a preprint [here] entitled Équations elliptiques du second ordre à coefficients discontinus published in Séminaire Jean Leray in 1963-1964.
Suppose is a non-negative non-decreasing function satisfying
for any where are positive given constants.
- If , it holds
- If , one has
- If and , then
We shall only prove the case of .
Proof. Since is given, we can consider the following quantity
Then using the assumption for and it holds by
We now prove by induction the following
Indeed, for , the inequality is trivial. Suppose it holds for , we are going to prove for . We now have from the recursive equation that
Now sending with the fact that we can complete our proof.
For the case of , we just consider
then it holds
For the latter case, we shall consider the following auxiliary function
Then it holds
For the complete proof, we refer the reader to Stampacchia’s paper for details.