I am interested in solutions to the Monge-Ampere equation for a smooth function $h(x,y)$ of two variables(though I suppose I could try to make do with $C^2$ solutions). The equation is:
$$det(Hess(h(x,y))=0 $$
Here $det$ denotes the determinant function. Geometrically this says that we have a smooth developable surface $(x,y,h(x,y))$ of constant Gaussian curvature.
I want this solution to be defined a half-plane where $x > R $ for some $ R >> 0$. And I want it to satisfy the following properties.
$$h(x,y)=x \quad when \quad |y| < \epsilon $$ $$ h(x,y)=\sqrt{x^2 + y^2} \quad when \quad |y| > 2\epsilon $$
Do such solutions exist?