Ngô Quốc Anh

June 19, 2010

Existence of global super-solutions of the Lichnerowicz equations

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 19:16

This entry devotes the existence of global super-solutions to the Lichnerowicz equations in the study of the Einstein equations in general relativity. This terminology plays an important role in the study of non-CMC case of the Einstein equations in vacuum case.

This terminology first introduced in 2009 by a paper due to  M. Holst, G. Nagy and G. Tsogtgerel published in Comm. Math. Phys. [here]. In that paper, the solvability comes from the existence of both global sub- and global super-solutions together with some new fixed-point arguments which we have already discussed before [here]. Maxwell recently got a significant result by relaxing the existence of global sub-solutions [here] so that the existence of global super-solutions is enough to guarantee the solvability of the Einstein equations in non-CMC case.

In the vacuum case, the classification depends on the sign of the Yamabe invariants.

In the conformal method, in three dimensions the study of the Einstein equations becomes the study of existence of solution (\varphi, W) to a coupled system:

\displaystyle -8\Delta \varphi + R\varphi=-\frac{2}{3}\tau^2\varphi^5+|\sigma+\mathbb LW|^2\varphi^{-7}

and

\displaystyle {\rm div} \mathbb LW=\frac{2}{3}\varphi^6d\tau.

The first equation is usually called the Lichnerowicz equation

\displaystyle -8\Delta \varphi + R\varphi=-\frac{2}{3}\tau^2\varphi^5+|\beta|^2\varphi^{-7}

where \beta is a symmetric (0,2)-tensor.

Definition. We say \varphi_+ is a super-solution of the Lichnerowicz equation if

\displaystyle -8\Delta \varphi + R\varphi=-\frac{2}{3}\tau^2\varphi^5+|\beta|^2\varphi^{-7}.

Now we have

Definition. We say \varphi_+ is a global super-solution of the Lichnerowicz equation if whenever

0<\varphi \leqslant \varphi_+

then

\displaystyle -8\Delta \varphi + R\varphi=-\frac{2}{3}\tau^2\varphi^5+|\sigma+\mathbb LW_\varphi|^2\varphi^{-7}

where W_\varphi is a solution of second equation obtained from \varphi.

We are now in a position to derive main results.

The case of non-negative Yamabe invariants (far-from-CMC). This result was due to M. Holst et al.

Theorem. Suppose g \in W^{2,p} with p>3 and \mathcal Y_g>0, \tau \in W^{1,p} and \sigma \in W^{1,p}. If \|\sigma\|_\infty is sufficiently small, then there exists a global super-solution.

Proof. Pick \psi \in W_+^{2,p} such that the scalar curvature \hat R of \hat g=\psi^4g is strictly positive. We claim that if \varepsilon is sufficiently small, and if \|\sigma\|_\infty is additionally sufficiently small, then \varepsilon\psi is a global super-solution.

Suppose 0<\varphi\leqslant\varepsilon\psi, and let W_\varphi be the corresponding solution of the second equation. Note that

\displaystyle\begin{gathered} -8\Delta (\varepsilon\psi) + R(\varepsilon\psi)+\frac{2}{3}\tau^2(\varepsilon\psi)^5-|\sigma+\mathbb LW_\varphi|^2(\varepsilon\psi)^{-7}\\=\varepsilon\hat R\psi^5+\frac{2}{3}\tau^2(\varepsilon\psi)^5-|\sigma+\mathbb LW_\varphi|^2(\varepsilon\psi)^{-7}\\\geqslant\varepsilon\hat R\psi^5-2|\mathbb LW_\varphi|^2(\varepsilon\psi)^{-7}-2|\sigma|^2(\varepsilon\psi)^{-7}\end{gathered}

by the Cauchy-Schwarz inequality.

It is well-known that there exists a constant K_\tau such that

\displaystyle \|\mathbb LW_\varphi\|_\infty \leqslant K_\tau\|\varphi\|_\infty^6 \leqslant K_\tau \varepsilon^6\max(\psi)^6.

Hence

\displaystyle\begin{gathered}\varepsilon\hat R\psi^5-2|\mathbb LW_\varphi|^2(\varepsilon\psi)^{-7}-2|\sigma|^2(\varepsilon\psi)^{-7}\\\geqslant \varepsilon\min(\hat R)\min(\psi)^5 -2K_\tau^2\varepsilon^5\max(\psi)^{12}\min(\psi)^{-7}-2|\sigma|^2(\varepsilon\psi)^{-7}\hfill\\=\varepsilon 2K_\tau^2 \frac{\max(\psi)^{12}}{\min(\psi)^7}\left[ \frac{\min(\hat R)}{2K_\tau^2}\left( \frac{\min(\psi)}{\max(\psi)}\right)^{12}-\varepsilon^4\right]-2|\sigma|^2(\varepsilon\psi)^{-7}\hfill.\end{gathered}

Now pick \varepsilon so small that

\displaystyle\frac{\min(\hat R)}{2K_\tau^2}\left( \frac{\min(\psi)}{\max(\psi)}\right)^{12}-\varepsilon^4

is positive. It then follows that \varepsilon\psi is a global super-solution so long as \|\sigma\|_\infty is so small that the RHS of the above inequality remains positive.

The case of negative Yamabe invariants (near-by-CMC). This is my own result. I found this about several weeks ago.

Theorem. Suppose g \in W^{2,p} with p>3 and \mathcal Y_g<0, \tau \in W^{1,p} and \sigma \in W^{1,p}. If \|\sigma\|_\infty is sufficiently small and \tau is bounded away from zero, say |\tau|\geqslant\varepsilon_0>0, then there exists a global super-solution.

Proof. In order to deal with this case, we need a refined pointwise estimate for \mathbb LW_\varphi, precisely, we have

\|\mathbb LW_\varphi\|_\infty <C\max(\varphi)^6\max(\nabla \tau).

This result can be found [here]. We now pick \psi \in W_+^{2,p} such that the scalar curvature \hat R of \hat g=\psi^4g is constant which also satisfies

\displaystyle |\hat R| < \frac{1}{3}\varepsilon_0^2.

This is nothing but the Yamabe problem. We claim that if \|\sigma\|_\infty is additionally sufficiently small, then \psi is a global super-solution.

Indeed,

\displaystyle\begin{gathered} -8\Delta \psi + R\psi+\frac{2}{3}\tau^2\psi^5-|\sigma+\mathbb LW_\varphi|^2\psi^{-7}\\\qquad=\varepsilon\hat R\psi^5+\frac{2}{3}\tau^2\psi^5-|\sigma+\mathbb LW_\varphi|^2\psi^{-7}\hfill \\\qquad\geqslant\frac{1}{3}\tau^2\psi^5-2|\mathbb LW_\varphi|^2\psi^{-7}-2|\sigma|^2\psi^{-7}.\hfill\end{gathered}

Note that

\displaystyle \frac{1}{3}\tau^2\psi^5-2|\mathbb LW_\varphi|^2\psi^{-7} \geqslant \frac{1}{3}\tau^2\psi^5-2C^2\frac{\max(\psi)^{12}}{\min(\psi)^7}\max(\nabla \tau)^2.

So if \left|\frac{\nabla \tau}{\tau}\right| is sufficiently small, we have

\displaystyle \frac{1}{3}\tau^2\psi^5-2|\mathbb LW_\varphi|^2\psi^{-7} >0.

It then follows that \psi is a global super-solution so long as \|\sigma\|_\infty is so small that the RHS of the above inequality remains positive.

_\varphi

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