Let $D=\{(x,y):x^2+(y-2)^2<4\}$ with its boundary $\partial D$.Consider the boundary value problem $u_{xx}+u_{yy}=0$ in $D$ and $u(x,y)=x^2-y^2$ on $\partial D$. Then the $\max\{u(x,y):(x,y)\in D \cup \partial D \}$ is?
Now to approach this problem i know that the maximum value will occur at the boundary using maximum modulus principle. I am getting my maximum value as 0. Is it true? Please help me with this one.