4
$\begingroup$

I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.

Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, it contains second order time term $u_{tt}$ and the determination of Hessian of solution $\det(D_x^2 u)$ (we suppose that $u$ is convex, which means that the nonlinear operator is elliptic). Recently, I have read some papers concerning Hyperbolic mean curvature flow like the paper The hyperbolic mean curvature flow by K. Smoczyk and philippe G. LeFloch on JMPA. But does there exist some research like the above question, if yes, can you offer me some paper? Thank advance.

Improve this question
$\endgroup$
8
  • $\begingroup$ @Denis Serre, Thank you for your correct, professor! $\endgroup$
    – monotone operator
    Commented Dec 4, 2023 at 3:48
  • 2
    $\begingroup$ The question is why should we study this equation? The only hyperbolic PDEs studied extensively in geometric analysis are those in general relativity. These are already quite difficult to analyze, but people are motivated to do so by the physical significance of the equations. Why would we want to study this particular PDE? In general, nonlinear hyperbolic PDEs are much more difficult to study than noninear elliptic or parabolic PDEs. $\endgroup$
    – Deane Yang
    Commented Dec 5, 2023 at 18:31
  • $\begingroup$ Thank you,@DeaneYang, this equation maybe meaningless. When I meet hyperbolic mean curvature flow, I want to know whether there are equations like this form. $\endgroup$
    – monotone operator
    Commented Dec 6, 2023 at 3:34
  • $\begingroup$ Almost all PDE has its physical or geometrical meaning, my question just a connection in my mind. And, are there equations like $u_{t}=\det(D^{2}u)+\dots?$ $\endgroup$
    – monotone operator
    Commented Dec 6, 2023 at 3:38
  • $\begingroup$ That’s just not true. Most PDEs have no physical or geometric meaning. Only some of the ones you see in books and papers. There are many others you never see that are of no interest at all. The generic PDE is a terrible thing. $\endgroup$
    – Deane Yang
    Commented Dec 6, 2023 at 3:50

0

Reset to default

You must log in to answer this question.