Suppose $$0 = - \Delta u+u \quad \text{in } U$$ where $U$ is open connected bounded set and $u(x)$ is $C^2(U)$ and $C(\partial U)$, and that $\max_{\partial U} u > 0$. Prove that
- $\max_{\partial U} u = \max_{\bar U}$
- there does not exist $x_0 \in U$ such that $u(x_0) = \max_{\bar U}$.
Proof: obviously $\max _{\partial U} u \leq \max_{\bar U}$ since $\partial U$ is a subset. Now if the max at the closure was attained by a $x_0$ at the boundary obviously they are equal. Suppose $x_0$ is in $U$. Now for a local maximum I have
$$\Delta u \leq 0 \implies -\Delta u+u \geq u \implies 0 \geq \max_{\bar U}u$$
and since $\max _{\partial U} u > 0$ I get
$$\max_{\partial U} u > \max_{\bar U}$$
which is impossible so they must be equal and $x_0$ be at the boundary. What am I missing? I kind of proved both questions at the same time and something feels fishy. Also I have not used the connectivity of my set.