I am working through L.C.Evans' Partial Differential Equations -- the chapter on second-order elliptic equations.
I have got a general question on symmetric vs. non-symmetric elliptic operators. Consider an operator of the form
$\displaystyle Lu = \sum_{i,j}a^{ij} u_{,ij} + \sum_i b^i u_{,i} + cu$
In his book, Evans mainly treats the case of a symmetric highest order part (i.e. the coefficients for the second order derivatives form a symmetric matrix, $a^{ij}=a^{ji}$). The author frequently tells this restriction is 'without loss of generality'. However, I don't see what he means by this. Hence I wonder
- You can you generalize to the case of $(a^{ij})_{ij}$ being non-symmetric?
- How does this relate with the operator as a whole being symmetric?
It would be great help if you could clarify this picture.