The elliptic operator (in divergence form) in (Evans, and many texts) is defined as $$Lu=-D_j(a^{ij}D_i u)+b^i D_i u+cu$$ $D_i u$ denotes $u_{x_i}$,assuming the summation convention is understood.
While in some other texts, there is a plus sign instead of a minus in front of the leading term, i.e. $$Lu=D_j(a^{ij}D_i u)+b^i D_i u+cu.$$
Obviously among all authors, the sign plays no role. But why? Could anyone reconcile the difference of the two versions? The sign does not matter, why?
Just as what you did on your previous question on MSE, to obtain a weak formulation of the problem, we multiply the $Lu$ by test functions $v\in H_0^1$ and integrate by parts. Doing so for the first term, \begin{align*} \int_\Omega -D_j(a^{ij}D_iu)v & = \int_{\partial\Omega} -(a^{ij}D_iu)v - \int_{\Omega} -a^{ij}D_iuD_jv\\ & = \int_\Omega a^{ij}D_iuD_jv \tag{1} \end{align*} where the first term vanishes due to $v\in H_0^1$. You should be able to see that if we have $D_j(a^{ij}D_iu)$ instead, equation (1) will have an extra minus sign instead. This is what I meant by people wanting to have a plus sign after integrating by parts.