Dirichlet problem to the ball with boundary data $1-2y^2$.

Question

Let $\omega=\{(x,y):x^2+y^2<1\}$ be the open unit disk in $\mathbb R^2$ with the boundary $\delta\omega$.If $u(x,y)$ be the solution of Dirichlet problem

$$\begin{cases} u_{xx}+ u_{yy}=0 & \text{in} \ \omega \\ u(x,y)=1-2y^2 & \text{on the boundary.} \end{cases} $$

Then $u(1/2,0)=?$

1. $-1$
2. $-1/4$
3. $1$
4. $1/4$

I have no idea how to find the solution. Does there exist a simple way to find solution? please someone help.

Thanks.

Answer 1

Hint: On the boundary, $x^2 + y^2 = 1$, so

$$1 - 2 y^2 = x^2 + y^2 - 2y ^2= x^2 - y^2.$$

What can you say about the function $x^2 -y^2$?