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.

(more…)

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