Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$. One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.
Let $U$ be a non-empty bounded open set of $\mathbb{R}^n$ ($n \geqslant 2$) and $f : U \rightarrow \mathbb{R}$ of class $\mathcal{C}^2$. Let $f$ be a function continuous on $\bar{U}$, of class $\mathcal{C}^2$ on $U$, and such that for all $x \in U$, $\Delta f(x) > 0$. Show that $x_0 \in \partial U$ and deduce that $\forall x \in U, f(x) < \sup_{y \in \partial U} f(y)$.
One may assume by contradiction that $x_0 \in U$, justify that there exists $i \in \llbracket 1, n \rrbracket$ such that $\frac{\partial^2 f}{\partial x_i^2}(x_0) > 0$, and consider the function $\varphi$ defined, for $t$ real, by $\varphi(t) = f(x_0 + t e_i)$, where $e_i$ denotes the $i$-th vector of the canonical basis of $\mathbb{R}^n$.