On the solution of a Monge-Ampere type non-linear partial differential equation

Question

In my research problem, I'm arrived at the following simple looking but highly non-linear pde which is related to the von Karman equations for plates with incompatible elastic strain (http://rspa.royalsocietypublishing.org/content/467/2126/402).

A sufficiently smooth (possibly analytic) function $w:X\to\mathbb{R}$ is given where $X$ is a simply connected bounded set in $\mathbb{R}^2$. Consider the partial differential equation

$$[\zeta,\zeta]=2[\zeta,w]$$

where $[f,g]:=f_{,xx} g_{,yy} + f_{,yy} g_{,xx} - 2 f_{,xy} g_{,xy}$.

Is anything known about the solution $\zeta:X\to\mathbb{R}$, in any category, of the above equation? Any reference would be appreciated!

Answer 1

When you write the solution, you must have some other conditions in mind, since one generally does not have unique solutions. For example $\zeta = 0$ and $\zeta = 2 w$ both satisfy your equation, so there is no uniqueness without further assumptions, such as boundardy conditions or initial conditions.

If one writes $\zeta = w + f$ where $f$ is a new unknown, then the equation becomes $$ [f,f] = [w,w], $$ which is a standard Monge-Ampère equation for $f$: $$ f_{xx}\,f_{yy}-{f_{xy}}^2 = w_{xx}\,w_{yy}-{w_{xy}}^2. $$ This equation is known to be elliptic if $[w,w]>0$ (in which case, it's appropriate to specify boundary values for $f$) and hyperbolic if $[w,w]<0$ (in which case, it is appropriate to specify initial values for $f$ in some appropriate sense).

In both cases, the properties of the global solutions depend on the shape of the domain $X$. For example, in the elliptic case, it would be appropriate to assume that $X$ is convex with a reasonably regular boundary. I believe that you might find some basic information on this in Gilbarg and Trudinger.

At places where $[w,w]$ vanishes, it is degenerate, and even local solvability is doubtful without knowing more about $w$.

There are, of course, many classic works on this equation, and a search on the relevant terms should turn up a wealth of information.