Partial differential equations with related functions Suppose we have $f(x, y) \geq 0$ and $g(x, y) \leq 0$ continuous, such that $\frac{d^2f}{dx^2} = \frac{d^2g}{dy^2}$. Can you show that $\frac{df}{dx} = 0$ or provide a counterexample?
EDIT: $f, g: \mathbb{R}^2 \rightarrow \mathbb{R}$

By chosing the domain $\mathbb{R}_+ \times \mathbb{R}_+$ and setting:
$$
f(x,y)=xy \geq 0\\
g(x,y)=-xy \leq 0
$$
Then we have 
$$
f_{xx}=0=g_{yy}
$$
But 
$$
f_x=y \neq 0
$$
If you want an answer for $\mathbb{R}^2$ as the "whole space", just specifiy it in your question. But this answer just came to my mind while reading your question.
