Orthogonal Level Sets and a generalization of harmonic functions Forgive my ignorance of differential equations and analysis.  I was playing around with orthogonal level curves of real valued functions in the plane, and realized this is one way a person could be led to harmonic functions:
The function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ has gradient
$$ \nabla f(x,y) = <f_x,f_y> $$
If we wanted a function $g$ so that the level sets of $g$ are orthogonal to those of $f$ then we can just take their gradients to be orthogonal.  The easiest way to do that is to find a $g$ so that
$$\nabla g = <-f_y,f_x> $$
But, any conservative vector field $<P,Q>$ must satisfy $P_y = Q_x$, which leads to the condition that
$$-f_{yy} = f_{xx}$$
i.e. that $\Delta f$, the Laplacian, must be zero.
Now, suppose that we didn't assume that $\nabla g$ were not exactly $<-f_y,f_x>$, but instead some arbitrary multiple at each point, so that the level sets are still orthogonal.  Does this give you a more general class of functions? Then the condition becomes
$$\nabla g = \lambda(x,y) <-f_y,f_x>$$
for some positive function $\lambda$.  Then, by the product rule, and using the same $P_y = Q_x$ condition above, we get that 
$$-\lambda_y f_y -\lambda f_{yy} = \lambda_x f_x + \lambda f_{xx}$$ 
Or in another form
$$\lambda \Delta f +\nabla \lambda \cdot \nabla f = 0$$
Dividing through by $\lambda$ and subtracting over, one could also write this as
$$\Delta f = \nabla (-log(\lambda)) \cdot \nabla f$$
So the question becomes, for a function $f$, does there exist some function $h$ so that 
$$\Delta f = \nabla(f) \cdot \nabla(h)?$$
Clearly, harmonic functions satisfy this with $\lambda = 1$ and $h = 0$.  Are there functions that satisfy this that aren't harmonic? Is this a well studied class of functions?  Again, forgive me as this might be a stupid question.
 A: These are solutions of a known class of partial differential equations: isotropic-coefficient elliptic PDE in divergence form. Indeed, the condition 
$$\nabla g = \lambda(x,y) \left<-f_y,f_x\right>$$
says that the field $\lambda(x,y) \left<-f_y,f_x\right>$ is conservative. In two dimensions, the condition of $\langle \alpha,\beta \rangle$ being  conservative  is equivalent to $\langle -\beta,\alpha \rangle$ having zero divergence. So, the property can be expressed as 
$$
\operatorname{div} (\lambda(x,y) \nabla f)=0 \tag{1}$$
which is the sort of equations I mentioned. 
A simple example of a non-harmonic function of the above kind is $f(x,y)=e^x$,  which satisfies (1) with $\lambda(x,y)=e^{-x}$. (Actually,  ellipticity requires that $\lambda$ be   pinched between positive constants, which fails here, though it holds locally. But you didn't ask about ellipticity bounds.)
A relatively recent paper by K. Astala, D. Faraco and L. Székelyhidi, Convex Integration and the $L^p$-theory of Elliptic Equations, deals with such equations and has pointers to older literature.
