Timeline for Reference on a Monge-Ampère-like equation
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 27, 2013 at 4:39 | comment | added | Jean-Marc Schlenker | Thanks Deane! Somehow if this system is not known this makes the question more interesting. We'll see what comes out of it... | |
| Nov 26, 2013 at 20:59 | comment | added | Deane Yang | Jean-Marc, I'm pretty confident that under some reasonable hypotheses, a sequence of smooth solutions would have a subsequence that converges to a smooth solution. I have never seen this equation before. It is a 2-by-2 system with the Monge-Ampère equation coupled to another elliptic PDE. Systems of 2nd order elliptic PDE's are rather rare, especially one involving the Monge-Ampère equation. | |
| Nov 26, 2013 at 13:45 | comment | added | Jean-Marc Schlenker | Deane (continuation): I hope we can get some results in this direction, but this PDE seems so simple and natural that we were wondering whether anybody had already thought about it. It would be surprising if it weren't the case?! | |
| Nov 26, 2013 at 13:44 | comment | added | Jean-Marc Schlenker | Deane: this problem comes up as a equivalent form of a problem we're studying (with Bonsante and Mondello) on some special, and I think interesting, maps between hyperbolic surfaces and hyperbolic 3-manifolds. (Actually to get the whole flavor we would need to look for functions invariant under a surface group acting on the disk, but let's forget about this.) In our context we can prove uniqueness and also existence of a "weak" solution. So what we're really interested in is knowing whether (under some hypotheses) a sequence of smooth solutions has a limit which is also smooth. | |
| Nov 26, 2013 at 13:32 | comment | added | Deane Yang | Proving global (rather than perturbed) results is a bit more challenging, but, as Willie says, there is a lot of existing technology for Monge-Ampère equations that can be applied to the real part and technology for linear elliptic PDE's applied to the imaginary part that can be combined to study the equation in greater depth. | |
| Nov 26, 2013 at 13:31 | comment | added | Deane Yang | Moreover, if you start with a sufficiently smooth solution, say, $$ u = \frac{1}{2}(x^2 + y^2) \text{ and }v = 0, $$ on a the unit disk, then you can use the implicit function theorem and the solution to the linear Dirichlet problem to show the existence of nearby solutions with perturbed boundary data. This would also show uniqueness of the Dirichlet problem for boundary data near the initial boundary data. | |
| Nov 26, 2013 at 13:31 | comment | added | Deane Yang | Could you say more about what you're looking for? As Willie points out, if $u$ and $v$ are assumed to have positive definite Hessians, this is a second order nonlinear elliptic system. This implies local existence of solutions. And there are minimal regularity conditions on $u$ and $v$ that imply $u$ and $v$ are smooth. | |
| Nov 22, 2013 at 16:12 | comment | added | Willie Wong | If the real part is strictly convex, then the imaginary part solves a linear elliptic equation with variable coefficients. If we consider the imaginary part fixed, the real part in fact solves a Monge Ampere equation. So playing a bit with the standard machinery would probably give you already some reasonable regularity results. | |
| Nov 22, 2013 at 14:04 | comment | added | Jean-Marc Schlenker | Thanks Willie Wong for those comments -- those remarks are actually quite interesting. In our case we know that $Re(w)$ has to be convex (or strictly convex), and we believe this should be strongly related to the (possible) regularity of $w$. | |
| Nov 22, 2013 at 13:57 | comment | added | Willie Wong | Come to think of it, isn't your equation solved by any function $w = u + i xy$ where $u = u(y)$? This already shows that the regularity can be as bad as your notion of solution (classical/weak/etc.) allows. | |
| Nov 22, 2013 at 13:53 | comment | added | Willie Wong | If you don't put additional constraints on $w$, regularity could potentially be bad. You can check that $w = i xy$ is a solution to your equation, and if you linearise the equation around it, the equation looks like $xy w_{xy} = 0$ which is, where it is not degenerate, hyperbolic. So I wouldn't hold up too much hope for a general regularity theory. | |
| Nov 22, 2013 at 9:51 | history | asked | Jean-Marc Schlenker | CC BY-SA 3.0 |