In this note, we shall prove the following
In this note, we shall prove the following
In this short note, we try to calculate conformal changes of Christoffel symbols that we have touched in the previous note [here]. Let us pick two Riemannian metrics and on a manifold sitting in the same conformal class, that is,
where is a smooth function on .
Let us recall that Christoffel symbols are determined by
In mathematical analysis, Ekeland’s variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exists nearly optimal solutions to some optimization problems. Ekeland’s variational principle can be used when the lower level set of a minimization problems is not compact, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland’s principle relies on the completeness of the metric space. Ekeland’s principle leads to a quick proof of the Caristi fixed point theorem.
Theorem (Ekeland’s variational principle). Let be a complete metric space, and let be a lower semicontinuous functional on that is bounded below and not identically equal to . Fix and a point such that
Then there exists a point such that
- and for all , .
Let us consider the Yamabe equation in the null case, that is
where is a compact manifold of dimension without boundary. We assume that is a smooth positive solution.
Since the manifold is compact without the boundary, the most simple result is
by integrating both sides of the equation. Now we prove that
Indeed, multiplying both sides of the PDE with and integrating over , one obtains
In the calculus of variations, a branch of mathematics, a Nehari manifold is a manifold of functions, whose definition is motivated by the work of Zeev Nehari (1960, 1961). It is a differential manifold associated to the Dirichlet problem for the semilinear elliptic partial differential equation
Here is the Laplacian on a bounded domain in .
There are infinitely many solutions to this problem. Solutions are precisely the critical points for the energy functional
on the Sobolev space . The Nehari manifold is defined to be the set of such that
Solutions to the original variational problem that lie in the Nehari manifold are (constrained) minimizers of the energy, and so direct methods in the calculus of variations can be brought to bear.
More generally, given a suitable functional J, the associated Nehari manifold is defined as the set of functions u in an appropriate function space for which
Here is the functional derivative of .
The Yamabe problem has been discussed here. Basically, starting from a metric , for a given constant Yamabe wanted to show there always exists a positive function such that the scalar curvature of metric defined to be equals . In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)
where the scalar curvature of metric . It is not hard to see that in the negative and null cases, two solutions of the Yamabe equation with are proportional. Let us prove the following
Theorem. In the negative and null cases, two solutions of the Yamabe equation with are constant .
We first fix a background metric and let be a solution to the Yamabe equation with . That is, the scalar curvature of metric equals .
We now consider a new metric still sitting in the same conformal class of with conformal factor such that its scalar curvature equals . In other words, is also a solution of the Yamabe equation with . Keep in mind
We have two cases.
Notice that the uniqueness result in the positive case is no longer true in general. Due to Obata, this is true for Einstein manifolds, that is, when the Ricci curvature and metric are proportional.
Let us prove following interesting identity between and
for any function .
For any function , using the formula
Since is a funtion, we know that
then we obtain
Now is a vector, we need to use the Riemmanian curvature tensor, which measures the difference between and .
Notice that there was an extra term involving Lie bracket. Fortunately, that term vanishes. Thus
The proof follows.
Let us do compare with respect to metric and with respect to Euclidean metric. For simplicity, let us try with function . Obviously,
Next, by definition of we obtain
So the question is why do they equal?
The purpose of this note is to introduce the squeezing method. As an application, we prove some Liouville type theorem for solutions to logistic equations. Here the text is adapted from a paper by Du and Ma published in J. London Math. Soc. in 2001 [here].
Let prove the following theorem.
Theorem. Let and be a non-negative stationary solution of
Then must be a constant.
The basic ingredients in the proof consist of the following three lemmas. But first of all, let us denote by a (uniformly) second order elliptic operator satisfying the PDE
with smooth, and
for some positive constants , .
Recently, I have read a paper entitled “Classification of solutions of a critical Hardy-Sobolev operator” published in J. Differential Equations [here].
In that paper, the authors try to classify all positive solutions for the following equation
where , and .
The main result of the paper can be formulated as follows
Theorem. Let be the function given by
Then is a solution to the equation above if and only if
for some and some .
First, they use the method of moving planes to prove the cylindrically symmetric of solutions. Thanks to this symmetry, the equation reduces to an elliptic equation in the positive cone in which eventually leads to a complete identification of all the solutions of the equation. We skip the detailed discussion here and refer the reader to the original paper.
This result has recently been generalized by Cao and Li.