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 to . In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)
where the scalar curvature of metric .
Clearly, one is also interested in the regularity of solutions to the Yamabe equation. We quote here the result due to N. Trudinger published in Ann. Scuola Norm. Sup. Pisa (3) in 1968 when he was a gradutate student. That paper can be found here.
Let us start with a result from elliptic theory.
Lemma. Let be a weak, non-negative solution in of the linear equation
where with . Then is positive and bounded and we have the estimates
where is constant depending only on and the function .
Yamabe’s approach. Yamabe tried to solve the subcritical case first. Precisely, he tried to solve the following equation
where . It is clear to see that the above equation can be rewritten as the following
Note that the function is of class where . Then once we can prove the existence of weak solution, this solution is positive and bounded. Since , elliptic regularity theory then guarantees that . By a standard bootstrap argument, one can show that . Repeat this process, eventually one has . At the final stage, Yamabe tried to let , by a compact embedding which is not true, we claimed that , solution to the above equation, goes to , the solution we need.
Trudinger’s approach. Trudinger tried to fix this error by assuming the quantity , called the mean scalar curvature of metric , is sufficiently small.
A Trudinger’s Regularity Theorem. This is the main part of this entry.
Theorem (Trudinger). Let be a solution of the Yamabe equation. Then .
Proof. Clearly, the function satisfies
for all test function . We choose an appropriate test function similarly to a method of Serrin published in Acta. Math. in 1965.
Define and for a fixed defined the functions
The function is a uniformly Lipschitz continuous function of and hence belongs to . Likewuse . Observe also that and vanish for negative and that
Let us now substitute in the variational equation test function where is an arbitrary, non-negative function. The result is
for some constant . Clearly, one can estimate further
Let us take now to have compact support in a coordinate patch of . The integrals above may then be replaced by integrals over a sphere in of radius where . We choose so that
Then applying the Holder and Sobolev inequalities, we get
We choose so that . Hence we may let to obtain the estimate
Let denote the sphere concentric to of radius and choose on , on . Then we obtain
Replacing by we have a full estimate for and employing a partition of unity clearly provides a global estimate
for some where will also depend on the local -norm of . The boundedness of and subsequently its smoothness are now consequences of the lemma above. The proof is complete.