Here is my question: Let $f(x,y)$ be a function that has continuous second-order partial derivatives on the unit disc $D$, and $$ \frac{\partial^2f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}=e^{-(x^2+y^2)} $$
prove $$\iint_D(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y})dxdy=\frac{\pi}{2e}.$$
Then I Use polar coordinate substitution $$\iint_D(x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y})dxdy=\int_0^1rdr\int_0^{2\pi}(r\cos {\theta}f_x+r\sin\theta f_y )d\theta.$$
Use the chain rule, $$f_x=\frac{\partial f}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial x},\\ f_y=\frac{\partial f}{\partial r}\frac{\partial r}{\partial y}+\frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial y}. $$ But I don't know how to relate it with $\frac{\partial^2f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}$. And if use polar coordinate substitution to $\frac{\partial^2f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}$, This would involve a lot of calculations, but the problem should not be that complicated.
Can anyone help me ? Thanks in advance!