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

together with

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 andthen where

*Proof of theorem*. Keep in mind from the Rellich-Kondrachov theorem, the following embedding

holds for any . So if there is some satisfying

such that

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

where

.

If then we are done. If by the Rellich-Kondrachov theorem, we can find such . Otherwise,

where

.

We need to compare and . Actually, we need to show that

.

Indeed,

which implies

.

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.

## Leave a Reply