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 equationwhere 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

and

where .

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

and hence

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

and hence

.

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.

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