The pde


$$ Pf(x)=\sum \_{i,j=1}^{n}a\_{ij}(x){\frac {\partial ^{2}f}{\partial x\_{i}\,\partial x\_{j}}}(x)+\sum \_{i=1}^{n}b\_{i}(x){\frac {\partial f}{\partial x\_{i}}}(x)+c(x)f(x),$$
is said to be [semi-elliptic](https://en.wikipedia.org/wiki/Semi-elliptic_operator) when the matrix $(a\_{ij}(x))$ is *positive semidefinite* for all x.



> 
> Q1: What are some comprehensive references on them?
> 
> 
> 


They are not treated in Evans, and the reference in [wikipedia](https://en.wikipedia.org/wiki/Semi-elliptic_operator) is Oksendal's SPDEs book, which actually looks at the case $c\equiv 0$ and doesn't cover the same results as Evans.



Semi-elliptic pdes reference
I would have guessed that many do not pass through. One of the most basic things fails because for a divergence form operator $D\_i(a\_{ij}(x)D\_ju)$ you do not have
$$
\int |Du|^2 \leq \int a\_{ij}D\_iuD\_ju
$$
which is the basis of an energy estimate.


