Under what conditions is it possible, using a suitable change of variables, to eliminate 1st order terms in an elliptic partial differential equation, so that the equation involves the 2nd derivatives, the dependent variable, and independent terms only?


To be concrete, consider the elliptic equation $-\Delta u + \sum\_i \frac{d u}{dx^i} a^i + f(x)=0$. 


If the $a^i$ are constant, define $u(x) = v(x) e^{\frac{1}{2}\sum\_j a^j x^j}$ and obtain 


$-\Delta v - \frac{1}{4} v \sum\_i a^i a^i + f(x)e^{-\frac{1}{2}\sum\_j a^j x^j}=0$, an elliptic equation without 1st order terms.


If the $a^i$ are not constant or if the equation is quasilinear, the problem is harder. It can be approached using contact transformations and Cartan's method of equivalence, but I am not aware of results.



Eliminating 1st order terms in elliptic partial differential equation
The necessary and sufficient conditions for transforming one second-order differential operator of the type you are interested in into another are given by the so-called Cotton theorem, see Theorem 1 in [this paper](http://www.atlantis-press.com/php/download_paper.php?id=952) by Finkel and Kamran. In your case, you want the linear part of the transformed operator to vanish ($\tilde{\mathcal{A}}=0$ in the notation of the above paper), whence you can readily extract the conditions you ask for. Roughly speaking, in order to have no first-order terms in your transformed operator, the linear part of your original operator should be "pure gauge up to a change of independent variables".


