Integration by part in dimension 2

Question

I have an operator $\mathcal{L}$ defined for functions $u(x,y)$ on $\mathbb{R}^2$ $$\mathcal{L}:=x\partial_{xx}+\sqrt{xy}\partial_{xy}+\frac12y\partial_{yy}+2\rho\partial_x+\rho\partial_y.$$ I want to perform an integration by part for the integral $$ I:=\int\int_{x,y>0}\mathcal{L}u\cdot f dxdy $$ where $f$ is also a function on $\mathbb{R}^2$.I want to find an adjoint operator $\mathcal{L}^*$ such that $$ I=\int\int_{x,y>0}u\cdot\mathcal{L}^*f dxdy. $$ I guess it is a quite standard operation. I know it is related somehow to Stoke's theorem, but cannot figure out how to do this. Can anyone please give some hint on how to do this, or give me some reference? Thank you!