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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 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 is Oksendal's SPDEs book, which actually looks at the case $c\equiv 0$ and doesn't cover the same results as Evans.

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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.

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