Liouville's equation

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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 $u = 2φ$ and $Λ$ 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 $g = exp(φ) g 0$ 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.

References

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.

Liouville's equation

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