“Consider the PDE $u=u_{xx}+u_{yy}$ in dimension 2, where $u:\mathbb{D}\to\mathbb{R}$ is a scalar function on the unit disk $\mathbb{D}=\{(x,y)|x^2+y^2\leq1\}$. Show that if $u$ is a solution that is identically zero on the boundary $\partial\mathbb{D}=\{(x,y)|x^2+y^2=1\}$, then $u$ is identically zero on the whole disk. Assume $u$ is continuous.”
I said that $u=u_{xx}+u_{yy}=\nabla^2u$, so I thought about what that would mean. If $u$ wasn’t constant on $\mathbb{D}$, then at any local maximum of $u$, $\nabla^2u$ is at a local minimum, and vice versa. However, since $u=\nabla^2u$, I thought that this would be a contradiction, and therefore, $u$ must be constant and identically zero on $\mathbb{D}$.
The feedback I received said, “there is no a priori contradiction when one side of an inequality is at a maximum and the other side is a minimum. In other words $u$ and $u_{xx}+u_{yy}$ could just be two different functions doing there own things.”
Can someone explain why this is, and where my reasoning breaks down?