In this note, we discuss an useful lemma and its beautiful proof given by Brezis and Cabré in a paper published in Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. in 1998. The full article can be freely downloaded from here. Before saying anything further, let us state the lemma.
Lemma 3.2. Suppose that belongs to . Let be the solution of
where is a constant depending only on .
This type of estimate frequently uses in the literature. We now show the proof of the lemma.
The proof basically consists of two steps:
Step 1. For any compact subset , we show
for any where the constant depends only on and . This and the fact that stays away from the boundary of allow us to have the desired estimate. To do so, let
and take balls of radius such that
This can be done since is compact. Now, instead of proving our estimate in the whole , we shall prove our estimate for each ball . To do that, let be the solutions of
where denotes the characteristic function of . The Hopf boundary lemma implies that there is a constant such that
for all and all . Let now and take a ball containing . Then
and since in , we conclude
Step 2. Fix a smooth compact subset , we need to prove
for any . Let be the solution of
The Hopf boundary lemma now gives
for any . Since is superharmonic and on , the maximum principle implies
for any . The proof is now complete.