Let us now consider a very important technique in PDEs called the bootstrap method or bootstrap argument. Actually, this method is a part of the proof of the Schauder estimates, etc. I will follow the recent paper due to Li, Strohmer and Wang published in Proc. Amer. Math. Soc. in 2009.
The equation considered here is the following
where is a bounded domain. We also assume
Theorem. If is a solution to the PDE for some then .
In order to run the bootstrap argument, we need the following auxiliary result
Lemma. Suppose with and
Proof of theorem. Keep in mind from the Rellich-Kondrachov theorem, the following embedding
holds for any . So if there is some satisfying
we then see that
From the PDE and our lemma above we get
for any . Choosing gives us the desired result. Therefore we only need to consider the case that
The idea of the bootstrap argument is to try to increase the regularity of , in this situation, our aim is to obtain such from the fact that . Indeed, firstly using our lemma we actually have
If then we are done. If by the Rellich-Kondrachov theorem, we can find such . Otherwise,
We need to compare and . Actually, we need to show that
In other words, we reach to the case where . If does not satisfy
we then replace by and repeat the above argument again until does satisfy the above inequality. The proof then follows.