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Problem : $-\Delta u=1-\sqrt{x^2-y^2}$ in $B_1(0)$; $u=0$ on $\partial B_1(0)$

I could use the Poisson integral formula : $$u(x)=\int\limits_{B_1(0)}G(x,y)\Delta u(y)dy$$ where $G$ is the Green's function for the unit ball. But the integral looks daunting.

On the other hand I could write $u(x,y)=X(x)Y(y)$ and then solve the eigenvalue problem $u_{xx}+u_{yy}=-\lambda u$ which leads me to $$\frac{X''}{X}+\frac{Y''}{Y}=-\lambda$$ or $$X''+\mu X=0, Y''+(\lambda-\mu)Y=0$$ If it were a rectangular region, I would have had nice boundary conditions. I also thought about using polar coordinates but didn't go through with it.

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