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: I think many of the basic elliptic results pass through, but it would be good to have a comprehensive reference.

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.

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.

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: I think many of the basic elliptic results pass through, but it would be good to have a comprehensive reference.

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.

Also I would appreciate any feedback on the particular semi-elliptic pde I am studying:

$$\frac{1}{2}u_{xx}+\frac{x}{x^{2}+y^{2}}u_{x}-\frac{y}{x^{2}+y^{2}}u_{x}+c\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}u=0$$

for $(x,y)\in \mathbb{R}\times \mathbb{R}_{>0}$ and boundary $u(x,0)=g(x)$.

Q2: In particular, if you've seen some similar pde elsewhere. It will be very interesting if it has a closed form solution, but please only give hints because I want to practice.

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: I think many of the basic elliptic results pass through, but it would be good to have a comprehensive reference.

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.