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The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix $$ \left\|\frac{\partial^2u}{\partial x^i\partial x^j}\right\|\, . $$ For instance, with $n=2$, one has $$ A+Bu_{xx}+Cu_{xt}+Du_{tt}+E(u_{xx}u_{tt}-u_{xt}^2)=0\, ,\quad\quad (^*) $$ where $A,B,C,D,E$ depend on $x,t,u,u_x,u_t$.

Let's stick with the case $n=2$.

DEFINITION: A second-order bidimensional PDE is of Monge-Ampère type iff it can be brought in the form $(^*)$ by means of a contact transformation of the space $\{ x,t,u,u_x,u_t \}$.

Now, my question.

QUESTION: Is there an example of a second-order bidimensional PDE, which is a polynomial of degree not greater than 2 in the second-order derivatives, and is NOT of Monge-Ampere type?

What kind of answer do I expect? It would be really great if someone pointed out some "famous" equation (in the sense that it enjoys some popularity, possibly amongst physicists), like $$ u_t=u_{xx}\, ,\quad\quad (^{**}) $$ and provided me with a very simple argument showing that $(^{**})$ cannot be brought in the form $(^{*})$. (I simply don't believe that all the equations from my question are of Monge-Ampère type - yet this feeling may be wrong.)

COMMENTS after R. Bryant answer

To be more precise, my question is about hypersurfaces in second-order jets spaces or, in a fibre-wise perspective, about hypersurfaces in the Lagrangian Grassmannian. Of course, Monge-Ampère equations form an invariant class, as they correspond to hyperplane sections. So, I'm interested to know what lies beyond the class of hyperplane sections: intuitively, it should be the class of "hyperquadric" sections, and Bryant's "cylindrical" example $$ u_{xx}^2+u_{tt}^2=1 $$ falls indeed in such a class. Has this class been studied? Is there any "famous" example (e.g., describing the geometry of some special surfaces, or modeling some physical phenomenon) of a "hyperquadric" section which is not a hyperplane section?

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Two points:

First, the equation ${u_{xx}}^2+{u_{yy}}^2-1 =0$ is not of Monge-Ampère type.

Second, the class of Monge-Ampère equations is preserved under contact transformations on the $1$-jet space, so if an equation is of Monge-Ampère type up to contact transformation, then it is a Monge-Ampère equation already.

More precisely, one should distinguish between the particular equation and the hypersurface that it defines in the $2$-jet space. For example, the equation ${u_{xx}}^3 -1 = 0$ is clearly not in Monge-Ampère form, but it defines the same locus in the $2$-jet space as $u_{xx} - 1 = 0$, which is. Properly speaking, it is the hypersurface that is or is not of Monge-Ampère type, not the particular equation one uses to define it.

Comments after the modification of the question:

It is, perhaps, better to think of the class of Monge-Ampère equations as the smallest contact-invariant class that contains the quasi-linear second order scalar equations. That way, one can see that, since the Euler-Lagrange equations of first-order functionals are quasi-linear second-order equations, it is not surprising that many of the equations studied by physicists and mathematicians are of Monge-Ampère form. (One might wonder why they aren't all just second-order quasi-linear. That's a story involving the natural appearance of contact transformations and equivalence in PDE that is too long to go into here.) Thus, it's not that surprising that most of the scalar second-order equations one encounters in "physically motivated" problems belong to the Monge-Ampère class.

This also happens frequently in differential geometry, since many (though by no means all) of the equations we study are variational and second order.

An interesting class that is properly larger than Monge-Ampère equations that has been studied in basic differential geometry is the class of Weingarten equations, i.e., equations for a surface that impose a relation between the principal curvatures of the surface.

Of course, some of these are of Monge-Ampère type, such as the so-called linear Weingarten equations, i.e., equations of the form $a + 2b H + c K = 0$ where $a$, $b$, and $c$ are constants and $H$ and $K$ are the mean and Gaussian curvature, respectively.

However, the general relation $F(H,K)=0$ is a second order scalar equation that is not of Monge-Ampère type if the curve that it describes in the $HK$-plane is not linear. (The equation is elliptic (resp. hyperbolic) if the expression ${F_H}^2+2HF_HF_K + K{F_K}^2$ is positive (resp. negative) along the curve $F(H,K)=0$.)

Such equations have been studied in the differential geometry literature by many people. For a study of these equations as PDE, one could consult the classic papers of Caffarelli, Nirenberg, and Spruck that appeared in the 1980s.

Further comments on generalizing the Monge-Ampère class

In a certain sense, what makes the Monge-Ampère class "natural" (besides the fact that it includes the class of Euler-Lagrange equations for first order functionals) is that it is invariant under contact transformations, which carry Monge-Ampère equations to (other) Monge-Ampère equations. By contrast, even in the case of $2$ independent variables, the class of all second order equations that are quadratic in the second derivatives is not invariant under contact transformations, so, in that sense, it is not really a natural class to study. There are other contact-invariant natural classes, but explaining what they are requires an excursion into conformal geometry.

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  • $\begingroup$ Of course you're right! I've been deliberately sloppy, since the question is not about a mathematical problem, but rather an attempt to collect ideas and references, and MO is the best channel to get them: googling "hyperquadric sections of Lagrangian Grassmannians" is pointless (anyway, I had my question fixed a little). Is this "cylindrical" equation of yours a mere academic example, or it has been given some attention before/elsewhere? $\endgroup$ – Giovanni Moreno Dec 8 '15 at 10:39
  • $\begingroup$ So, starting from the linear Weingarten equation $F_{lin}$ (which btw, you wrote with too many $K$'s), the simplest way I can modify it and kick it out of the Monge-Ampère class, is by adding $H^2$. This way, $F:=F_{lin}+H^2$ is still of degree $\leq 2$ in the second-order derivatives, is not Monge-Ampère, and it can be interpreted as the equation behind a certain geometry. That's all I needed - thanks for the explanation! $\endgroup$ – Giovanni Moreno Dec 8 '15 at 21:07
  • $\begingroup$ @GiovanniMoreno: I have fixed the typo in the linear Weingarten equation (thanks for pointing it out). Does this answer your general question, or do you need more discussion? $\endgroup$ – Robert Bryant Dec 8 '15 at 22:57
  • $\begingroup$ being a "general question", it will never be answered in a fully satisfactory way! Of course, you have provided a lot of useful hints, and I'm just leaving the question unanswered as a bait for people willing to help... Now I go over these papers you've suggested, and I will try to formulate a more precise problem involving one (or a class) of the simplest Weingarten equations which are not of Monge-Ampère type. If the problem will be posted on MO, it will mean that I was able to formulate it, but not to solve! $\endgroup$ – Giovanni Moreno Dec 9 '15 at 18:23

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