Ngô Quốc Anh

March 2, 2010

On the regularity of solutions to the Yamabe problem

Filed under: Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 21:12

The Yamabe problem has been discussed here. Basically, starting from a metric g, for a given constant R Yamabe wanted to show there always exists a positive function \varphi such that the scalar curvature of metric \overline g defined to be \varphi^\frac{4}{n-2}g equals to R. In terms of PDEs, the scalar curvature satisfies the equation (called Yamabe equation)

\displaystyle \frac{{4\left( {n - 1} \right)}} {{n - 2}}\Delta \varphi + R\varphi = \overline R {\varphi ^{\frac{{n + 2}} {{n - 2}}}}

where \overline R the scalar curvature of metric \overline g.

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 u \in W^{1,2}(M) be a weak, non-negative solution in Mof the linear equation

\displaystyle \Delta u +fu=0

where f \in L^r(M) with r > \frac{n}{2}. Then u is positive and bounded and we have the estimates

\displaystyle\mathop {\sup }\limits_M \left\{ {|u|,\frac{1}{{|u|}}} \right\} \leqslant C\left( {n,{{\left\| f \right\|}_{{L^r}}}} \right)

where C is constant depending only on n and the function f.

Yamabe’s approach. Yamabe tried to solve the subcritical case first. Precisely, he tried to solve the following equation

\displaystyle \frac{{4\left( {n - 1} \right)}}{{n - 2}}\Delta \varphi + R\varphi = \overline R{\varphi ^{q-1}}

where q<\frac{2n}{n-2} = :N. It is clear to see that the above equation can be rewritten as the following

\displaystyle\Delta \varphi + \underbrace {\frac{{n - 2}}{{4\left( {n - 1} \right)}}\left( {\overline R{\varphi ^{q - 2}} - R} \right)}_f\varphi = 0.

Note that the function f is of class L^r where r=\frac{N}{q-2} > \frac{n}{2}. Then once we can prove the existence of weak solution, this solution is positive and bounded. Since r>\frac{n}{2}, elliptic regularity theory then guarantees that \varphi\in C^2(\overline M). By a standard bootstrap argument, one can show that \varphi^{(2)} \in C^2(\overline M). Repeat this process, eventually one has \varphi \in C^\infty(M). At the final stage, Yamabe tried to let q \to N, by a compact embedding which is not true, we claimed that \varphi_q, solution to the above equation, goes to \varphi, the solution we need.

Trudinger’s approach. Trudinger tried to fix this error by assuming the quantity \int_M Rdv_g, called the mean scalar curvature of metric g, is sufficiently small.

A Trudinger’s Regularity Theorem. This is the main part of this entry.

Theorem (Trudinger). Let u \in W^{1,2}(M) be a solution of the Yamabe equation. Then u \in C^\infty(M).

Proof. Clearly, the function u satisfies

\displaystyle\int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}{g^{ij}}{u_i}{\xi _j} + Ru\xi } \right)d{v_g}} = \overline R \int_M {\left( {|u{|^{N - 1}}\xi d{v_g}} \right)}

for all test function \xi \in W^{1,2}(M). We choose an appropriate test function \xi similarly to a method of Serrin published in Acta. Math. in 1965.

Define \overline u=\sup(u,0) and for a fixed \beta>1 defined the functions

\displaystyle G(\overline u ) = \left\{ \begin{gathered} {\overline u ^\beta }\quad \text{ if } \quad\overline u \leqslant l, \hfill \\ {l^{q - 1}}\left( {q{l^{q - 1}}\overline u - (q - 1){l^q}} \right) \quad \text{ if } \quad \overline u > l \hfill \\ \end{gathered} \right.

and

\displaystyle F(\overline u ) = \left\{ \begin{gathered} {\overline u ^q}\quad \text{ if } \quad\overline u \leqslant l, \hfill \\ q{l^{q - 1}}\overline u - (q - 1){l^q}\quad \text{ if } \quad\overline u > l \hfill \\ \end{gathered} \right.

where q=\frac{\beta+1}{2}.

The function G(\overline u) is a uniformly Lipschitz continuous function of u and hence belongs to W^{1,2}(M). Likewuse F(\overline u). Observe also that G and F vanish for negative u and that

\displaystyle {\left( {F'(\overline u )} \right)^2} \leqslant qG'(\overline u ), \quad \overline u G(\overline u ) \leqslant {\left( {F(\overline u )} \right)^2}.

Let us now substitute in the variational equation test function \xi=\eta^2G(\overline u) where \eta is an arbitrary, non-negative C^1(M) function. The result is

\displaystyle\frac{{4(n - 1)}}{{n - 2}}\mu \int_M {{\eta ^2}G'(\overline u )u_j^2d{v_g}} \leqslant \int_M {\left( {\frac{{4(n - 1)}}{{n - 2}}\sup |{g^{ij}}|\eta |{\eta _i}| + \sup |R|u{\eta ^2} + \overline R {u^{N - 2}}{\eta ^2}} \right)Gd{v_g}}

and hence

\displaystyle\int_M {{\eta ^2}G'(\overline u )u_j^2d{v_g}} \leqslant C\int_M {\left( {\left( {{\eta ^2} + |{\eta _i}{|^2}} \right)\overline u G(\overline u ) + {\eta ^2}{{\overline u }^{N - 2}}{\eta ^2}\overline u G(\overline u )} \right)d{v_g}}

for some constant C. Clearly, one can estimate further

\displaystyle\int_M {{\eta ^2}{{\left( {F'(\overline u )} \right)}^2}u_j^2d{v_g}} \leqslant Cq\int_M {\left( {\left( {{\eta ^2} + |{\eta _i}{|^2}} \right){{\left( {F(\overline u )} \right)}^2} + {\eta ^2}{{\overline u }^{N - 2}}{\eta ^2}{{\left( {F(\overline u )} \right)}^2}} \right)d{v_g}} .

Let us take \eta now to have compact support in a coordinate patch of M. The integrals above may then be replaced by integrals over a sphere S_R in E^n of radius R where \eta=\eta(x) \in C_0^1(R). We choose R so that

\displaystyle\int_{{S_R}} {|u{|^N}d{v_g}} \leqslant \frac{1}{{2Cq}}.

Then applying the Holder and Sobolev inequalities, we get

\displaystyle {\left\| {\eta F} \right\|_{{L^N}(S)}} \leqslant Cq{\left\| {(\eta + {\eta _i})F} \right\|_{{L^2}(S)}} + \frac{1}{2}{\left\| {\eta F} \right\|_{{L^N}(S)}}

and hence

\displaystyle {\left\| {\eta F} \right\|_{{L^N}(S)}} \leqslant 2Cq{\left\| {(\eta + {\eta _i})F} \right\|_{{L^2}(S)}}.

We choose \beta \in \left( {1,\frac{{n + 2}}{{n - 2}}} \right) so that 2q<N. Hence we may let l \to \infty to obtain the estimate

\displaystyle {\left\| {\eta {{\overline u }^q}} \right\|_{{L^N}(S)}} \leqslant C{\left\| {(\eta + {\eta _i}){{\overline u }^q}} \right\|_{{L^2}(S)}}.

Let S_{\frac{R}{2}} denote the sphere concentric to S of radius \frac{R}{2} and choose \eta=1 on S_{\frac{R}{2}}, |\eta_i| \leq \frac{2}{R} on S_{\frac{R}{2}}. Then we obtain

\displaystyle {\left\| {{{\overline u }^q}} \right\|_{{L^N}(S)}} \leqslant C\left( {1 + \frac{1}{R}} \right).

Replacing u by -u we have a full estimate for u and employing a partition of unity clearly provides a global estimate

\displaystyle {\left\| u \right\|_{{L^r}(S)}} \leqslant C

for some r>N where C will also depend on the local L^N-norm of u. The boundedness of u and subsequently its smoothness are now consequences of the lemma above. The proof is complete.

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