This note is to concern a recent result by David G. Costa [here]. Here the statement

Theorem 1.1. For all and one haswhere . In addition, if , then

where the constant is sharp.

Here’s the proof.

This note is to concern a recent result by David G. Costa [here]. Here the statement

Theorem 1.1. For all and one haswhere . In addition, if , then

where the constant is sharp.

Here’s the proof.

In the literature, there is an inequality called the Alexandrov-Bol inequality which is frequently used in partial differential equations. Here we just recall its statement without any proof.

Theorem. Let be a good domain in . Assume be a positive function satisfying the elliptic inequalityin . Then it holds

where

and

.

An analytic proof was given by C. Bandle aroud 1975 when she assumed to be real analytic. The above version was due to Suzuki in an elegant paper published in the *Ann. Inst. H. Poincare* in 1992 [here]. The proof is mainly depended on the isoperimetric inequality for the flat Riemannian surfaces. We refer the reader to the paper by Suzuki for the proof.

We now prove the following result

Theorem. Let and be two smooth functions on satisfying.

Suppose that is bounded and also and

.

Then

.

Long time ago, we studied [here] the following fact

Suppose with . Define

.

Show that is finite for all and .

In this entry, from now on we continue to prove several useful results appearing in PDE. We shall prove the following

Theorem. Assume is a solution towith finite energy

.

Then

.

Let us consider the following equation

for and . In this entry, by using boothstrap argument, we show that

Theorem. If positive function solves the equation, then .

In the process of proving the result, we need the following result

Proposition. Let be a non-negative function and set.

For , there exist positive constants and depending only on and such that for any with , and satisfying

we have

.

My purpose is to derive some regularity result concerning the following integral equation

where is open and bounded and . To this purpose, in this entry we first consider the equation

for a suitable choice of .

**The case **. We will prove that for any . Indeed, up to a constant factor, the first derivative of are given by

.

From this formula,

.

By the intermediate value theorem, on the line from to , there exists some with

.

I just read the following result due to Loss-Sloane published in *J. Funct. Anal.* this year [here]. This is just a lemma in their paper that I found very interesting.

Let be any function in . Consider the inversion and set

.

Then and

.

*Proof*. For fixed consider the regions

and likewise,

.

Let us recall the following result

Theorem. Let be an open set and . If then for any , we havewhere over .

The above result has been proven in this entry. If we chose then we would have

.

If we now replace , we get

.

By differentiating with respect to , we arrive at

.

In this entry, we shall prove the foregoing identity is actually true. This is the second interesting formula I have mentioned before. The proof, of course again, is based on a clever choice of test function.

The classical Hardy inequality in , , is stated as follows

Theorem(Hardy’s inequality). Let with . Thenand

.

The constant is the best possible constant.

I suddenly found a very simple proof due to E. Mitidieri [here].

I found two interesting formulas related to co-area formula while reading some tricks done by Talenti regarding to the best constant of the Sobolev inequality. The first result is to derive a representation of

and the second result is to deal with differentiation of level sets. Having all these stuffs, I will derive a very short and beautiful proof concerning the lower bound of where , a positive solution to the following PDE

This proof I firstly learned from a paper published in *Duke Math. J.* in 1991 by W. Cheng and C. Li [here].

**Co-area formula**. Suppose that is an open set in , and is a real-valued Lipschitz function on . Then, for an integrable function

where is the -dimensional Hausdorff measure.

**The Sard theorem**. Let be , times continuously differentiable, where . Let be the critical set of , the set of points in at which the Jacobian matrix of has . Then has Lebesgue measure in .

The Sard theorem has some useful applications. For example, if the space of test functions where , then for almost every in the range of , we have that on the level set . Thus that level set will be an -dimensional surface. Furthermore

and

.

Theorem. Let be an open set and . If then for any , we havewhere over .