UNIFORM LOWER BOUND AND LIOUVILLE TYPE THEOREM FOR FRACTIONAL LICHNEROWICZ EQUATIONS

We study the fractional parabolic Lichnerowicz equation $$ \begin{align*} v_t+(-\Delta)^s v=v^{-p-2}-v^p \quad\mbox{in } \mathbb R^N\times\mathbb R \end{align*} $$ where $p>0$ and $ 0<s<1 $ . We establish a Liouville-type theorem for positive solutions in the case $p>1$ and give a uniform lower bound of positive solutions when $0<p\leq 1$ . In particular, when v is independent of the time variable, we obtain a similar result for the fractional elliptic Lichnerowicz equation $$ \begin{align*} (-\Delta)^s u=u^{-p-2}-u^p \quad\mbox{in }\mathbb R^N \end{align*} $$ with $p>0$ and $0<s<1$ . This extends the result of Brézis [‘Comments on two notes by L. Ma and X. Xu’, C. R. Math. Acad. Sci. Paris349(5–6) (2011), 269–271] to the fractional Laplacian.

Covariant diagonalization of the perfect fluid stress–energy tensor

We introduce new tetrads that manifestly and covariantly diagonalize the stress–energy tensor for a perfect fluid with vorticity at every spacetime point. This new tetrad can be applied to introduce simplification in the analysis of astrophysical relativistic problems where vorticity is present through the Carter–Lichnerowicz equation. We also discuss the origin of inertia in this special case from the standpoint of our new local tetrads.