For differentiable functions and , the following Picone’s identity is well known

The proof is very simple. For each partial derivative we have

which implies

Thus

The Picone identity is very useful. We shall address this later on.

For differentiable functions and , the following Picone’s identity is well known

The proof is very simple. For each partial derivative we have

which implies

Thus

The Picone identity is very useful. We shall address this later on.

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We consider the following PDE

.

By letting

via the potential theory, we has already proved that

As such, the analysis of turns out to be the core of the studying of solutions to our PDE. As in this entry, we showed that the following limit

exists for certain function . Not just the behavior at the infinity, as a question proposed also in that entry, we can control the decay rate of

i.e. we need the fact

for some positive constant where is a particular solution to

I do think this result is correct since it has been used once in a paper by X.X. Chen published in *Calc. Var. Partial Differential Equations *[here] but some idea is involved. I leave here as my own open question needed to be addressed in the future.

Let us continue the problem of prescribing Gaussian curvature. Our PDE reads as the follows

where is a compact manifold without the boundary. Today we show that if

then our PDE has unique solution.

Assume that and are solutions to the PDE, that is

By subtracting, we have

Multiplying both sides by , integrating over , and the using the integration by parts we arrive at

Since , it follows that

In particular, .

Let be a smooth and compact two dimensional Riemannian manifold. Let be a metric on with the corresponding Laplace-Beltrami operator and Gaussian curvature . Given a function on , can it be realized as the Gaussian curvature associated to the point-wise conformal

metric

To answer this question, it is equivalent to solve the following semi-linear elliptic equation

In this entry, we summarize some basic steps in order to simplify the above PDE. We first let , then our PDE becomes

Let be a solution of the following PDE

where

is nothing but the average of over . The solvability of the foregoing PDE comes from the fact that

We let . Then it is easy to verify that solves the following

Finally, letting

we get

or by renaming by

The advantage of this equation is that here is constant. To be precise, by the Gauss-Bonnet theorem, we have

where is the characteristic of .

In this entry, we prove the following interesting result

Let be a smooth function. Then

for each fixed.

For simplicity, let us write . Then

Last time, we discussed [here] Jacobi’s formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the differential of A. The formula is

.

A more useful formula is the following

.

Let us firstly reprove the Jacobi formula. Assuming is the cofactor matrix with respect to . It then holds

Therefore,

In this short note we present a result in a paper due to Edward M. Fan [here]. To be precise, we prove

Given a sphere with standard metric, if and , then must be a constant.

*Proof*. To prove the result, we shall use the following well-known formula

Therefore, the fact that is harmonic implies that

Since , multiplying both sides by , we get

Integrating both sides on using standard volume form, we get

Now integration by parts shows that

This in turn shows that

We conclude that must be a constant function.

In this short note, we prove the following identity

For any functions , it holds

The proof is elementary. By the product rule for the gradient, we know that

Thus

The term can be rewritten as follows

We then have

Keep in mind that

Therefore

Let us recall from this topic the following fact: Let be a compact Riemannian -manifold, and let and denote the Ricci tensor and the scalar curvature of , respectively. The so-called Paneitz operator acts on a smooth function on via

which plays a similar role as the Laplace operator in dimension two where is the de Rham differential. Associated to this operator is the notion of -curvature given by

Under the following conformal change

passing from to is easy through the following formula

This entry devotes an existence result for the following semilinear elliptic equation

in the whole space where .

Our aim is to apply the implicit function theorem. It is known in the literature that

Theorem(implicit function theorem). Let be Banach spaces. Let the mapping be continuously Fréchet differentiable.If

,

and

is a Banach space isomorphism from onto , then there exist neighborhoods of and of and a Frechet differentiable function such that

and if and only if , for all .

Let us now consider

.

Let us define

.

It is not hard to see that Fréchet derivative of at with respect to in the direction is given by

.

Since defines an isomorphism from to , it is clear to see that our PDE is solvable for small enough in the -norm.