A Poisson Equation: how to solve it? I wish to solve the Poisson Equation:$$u_{xx}+u_{yy}=1,$$with $u(x,y)$vanishing on $r=a$. 
I know that the final solution should be the sum of the homogenous part and the non-homogeneous part, which is $u(x,y)=u_1+u_2$, $u_1$ is the solution of $$u_{xx}+u_{yy}=0, u=0 \ on\ r=a,$$ and $u_2$ is the solution of $$u_{xx}+u_{yy}=1, u=0 \ on\ r=a. $$ I have no trouble to find $u_1$, but how to solve $u_2$? 
Any help, please. 
 A: How about starting with any solution $u_2$ of $u_{xx}+u_{yy}=1$, disregarding boundary conditions? You can think of solutions to that one without breaking a sweat. Now you have to find $u_1$ solving the homogeneous equation but with inhomogeneous boundary conditions, determined by the $u_2$ that you found in the first part.
A: You take the solution as two parts.
for the harmonic part,  we know $u_1$ is harmonic and vanishes at the boundary will give you the unique answer $u_1 = 0$
for the non-harmonic part, we can use Green's function method to get
$G(x-x_0,y-y_0)$, which satisfies
$\Delta G = \delta(x-x_0,y-y_0)$
$G(x-x_0,y-y_0)=0$ on boundary.
We know the fundamental solution is $\Gamma(x-x_0,y-y_0) = \dfrac{1}{4\pi}\ln((x-x_0)^2+(y-y_0)^2)$,
Then $G(x-x_0,y-y_0) = \Gamma(x-x_0,y-y_0)+ v(x,y,x_0,y_0)$
where $v$ satisfies
$\Delta v =0$
$v = f = -\Gamma(x-x_0,y-y_0)$ on boundary
Thus $v$ can be solved by using formula
$v = D\ast f$, where $D$ is Dirichlet kernel.
By integrating, I got $u_2 = \dfrac{1}{4}(x^2+y^2-a^2)$
