Minimal surface equation: Difference between revisions
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<center><math>\partial_\alpha [ (1 + \phi_\beta \phi^\beta )^{-1/2} \phi_\alpha ] = 0</math></center> | <center><math>\partial_\alpha [ (1 + \phi_\beta \phi^\beta )^{-1/2} \phi_\alpha ] = 0</math></center> | ||
where <math>\phi</math> is a scalar function on R^{n-1} | where <math>\phi</math> is a scalar function on <math>R^{n-1} \times R</math> (the graph of a surface in <math>R^n \times R</math> ). This is the Minkowski analogue of the minimal surface equation in Euclidean space, see [[Hp1994]]. | ||
* This is a [[QNLW|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. | * This is a [[QNLW|quasilinear wave equation]], and so LWP in <math>H^s</math> for <math>s > n/2 + 1</math> 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]]. | * GWP for small smooth compactly supported data is in [[Lb-p]]. | ||
Revision as of 20:46, 3 August 2006
The (hyperbolic) minimal surface equation takes the form
![{\displaystyle \partial _{\alpha }[(1+\phi _{\beta }\phi ^{\beta })^{-1/2}\phi _{\alpha }]=0}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a548f3cd428ef040d87057e77fa7cd92b9a19837)
where is a scalar function on (the graph of a surface in ). 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 for 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.
