The aim of this entry is to derive the -boundedness for a single solution of the following PDE

over a domain . This elegant result had been done by Brezis and Merle around 1991 published in *Comm. Partial Differential Equations* [here].

There are two possible cases.

**The case of bounded domain**. Let us assume a solution of the following PDE

where is a bounded domain and is a given function on .

Theorem. If and for some then .

Proof. It first follows from the Brezis-Meler inequality thatwhich by the Holder inequality gives

.

Therefore, if

while if

.

Thus, a standard -estimate argument from the elliptic theory implies that is bounded.

The conclusion of our theorem still holds for a solution of

with and for some .

**Remark**. It is worth noticing that a local version of the corollary mentioned in this entry still holds. Taking into account this result still gives a local version of the theorem above.

Theorem(local version). If and for some then .

**The case of the whole space**. Let us now turn to the case of . Our PDE is simply

.

In this circumstance, we get

Theorem. Assume . If and for some then .

Proof. Fix and split as with.

Let be the unit ball of radius centered at . We denote by various constants independent of . Let be the solution of

By the Brezis-Meler inequality,

and in particular

.

We also have

.

Let so that on . The mean value theorem for harmonic functions implies that

.

On the other hand we have

and since

we see that

.

Thus we find that

.

Finally we write

with

for some . Using once more the mean value theorem and standard elliptic estimates we deduce that

.

Since is independent of , we conclude that .

**Remark**. The mean value theorem for harmonic functions says that if is harmonic then

for any point . Thus

.

Taking sup both sides we arrive at

.

Congratulations for the beautiful website, full of amazing purposes and tricks!

In particular, this file about L-infinity boundness argument is been to me so useful during the reading of the elegante Brezis-Merle paper.

A dumb question: in the whole paper BM use a “standar elliptic estimate” and (exactly in the proof above) a “standard Lp estimate” that I don’t know.

May I have some info about or book references?

Thanks in advance!

Fab

Comment by Fab — December 6, 2010 @ 18:38

Hi Fab,

Thanks for your interest in my blog. Concerning to the standard estimate, I strongly recommend you to go through the book due to Gilbarg-Trudinger entitled “Elliptic partial differential equations of second order”.

Comment by Ngô Quốc Anh — December 7, 2010 @ 0:49

Thanks for your advice!

Comment by Fab — December 10, 2010 @ 20:56

However, if you use as boundary condition u<=0 instead of only u=0 you have no more modulus problem.

A suggestion: Why don't you post something on "the three alternative theorem" by Brezis Merle that is in the same elegant paper?

Fab

Comment by Fab — December 17, 2010 @ 22:35

Thank Fab, I will think about it.

Comment by Ngô Quốc Anh — December 17, 2010 @ 22:37

The comment above, of course, is refered to Brezis Merle Inequality, sorry!

Comment by Fab — December 17, 2010 @ 22:38

Sorry for my english mistake, refer is a transitive verb, so I should use only “is referred” without the “to”, isn’t right?

Comment by Fab — December 17, 2010 @ 22:44

Aha, you need to use “to”, so the latter was not correct but the first one.

Comment by Ngô Quốc Anh — December 17, 2010 @ 23:08