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.

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.