Liouville's equation

Liouville's equation

$\Box u=\exp(u)$

in $R^{1+1}$ first arose in the problem of prescribing scalar curvature on a surface. It can be explicitly solved as

$u=\log {\frac {8f'(x+t)g'(x-t)}{(f(x+t)+g(x-t))^{2}}},$

as was first observed by Liouville.

It is a limiting case of the sinh-gordon equation.

Standard energy methods give GWP in H^1.

Liouville equation turns out to be an equation for a Ricci soliton in $R^{2}$. This can be seen by noticing that the Ricci flow in this case take the very simple form

$\partial _{t}\phi =\exp(-2\phi )\triangle \phi .$

Then, a Ricci soliton is given by

$\triangle u=\Lambda \exp(u)$

after having set $u=3\phi$ and $\Lambda$ being a constant. We have used the fact that in dimension two, a set of isothermal coordinates always exists such that the Riemannian metric takes the simple form $g=\exp(\phi )g_{0}$ being $g_{0}$ the usual Euclidean metric. The equation for the Ricci soliton can be turned back to the original Liouville equation by a ${\frac {\pi }{2}}$ rotation of one of the coordinates in the complex plane.

See also

• The wikipedia entry for this equation
• A blog post on this equation by Terence Tao

References

1. J. Liouville, Sur l'equation aux differences partielles $\partial ^{2}\ln \lambda /\partial u\partial v\pm 2\lambda q^{2}=0$, J. Math. Pure Appl. 18(1853), 71--74.