Minimal surface equation

The (hyperbolic) minimal surface equation takes the form

$\partial _{\alpha }[(1+\phi _{\beta }\phi ^{\beta })^{-1/2}\phi _{\alpha }]=0$

where $\phi$ is a scalar function on $R^{n-1}\times R$ (the graph of a surface in $R^{n}\times R$ ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see Hp1994.

• This is a quasilinear wave equation, and so LWP in $H^{s}$ for $s>n/2+1$ follows from energy methods, with various improvements via Strichartz possible. However, it is likely that the special structure of this equation allows us to do better.
• GWP for small smooth compactly supported data is in Lb-p.