Maximum principle

Remarks

Proof idea for part 1: suppose that $L u > 0$ and $u$ had an interior maximum at $x_0$. Then $\nabla u(x_0) = 0$ and the Hessian of $u$ at $x_0$ would be negative semidefinite, which forces $L u(x_0) \le 0$. For $L u \ge 0$, apply this to $u + \varepsilon e^{\lambda x_1}$ with $\lambda$ large, and let $\varepsilon \to 0$. Part 2 follows from the Hopf lemma on the boundary normal derivative at a maximum point. Replacing $u$ by $-u$ gives the corresponding minimum principles for $L u \le 0$.

Example: every solution of Laplace's equation on a bounded open set takes its maximum and its minimum on the boundary. So a solution of the Dirichlet problem with boundary values between $0$ and $1$ lies between $0$ and $1$, and the Dirichlet problem has at most one solution.

The hypotheses are needed. Boundedness of $U$: on the half-plane $\set{x_1 > 0}$, the harmonic function $u = x_1$ vanishes on the boundary but is positive inside. The absence of a zeroth-order term with the wrong sign: on $(0, \pi)$, $u = \sin x$ satisfies $u'' + u = 0$, vanishes at both ends and equals $1$ at $\pi/2$. With a term $c(x) u$ and $c \le 0$, part 1 still holds in the form $\max_{\overline{U}} u \le \max_{\partial U} u^+$, where $u^+ = \max(u, 0)$.

Going deeper

For a student

The maximum principle is the main tool for uniqueness, comparison and bounds for second-order elliptic equations, without any integral estimates. It gives uniqueness for the Dirichlet problem, and comparison: if $L u \ge L v$ in $U$ and $u \le v$ on $\partial U$, then $u \le v$ in $U$. It also underlies Perron's method for the Dirichlet problem, Harnack inequalities and the theory of viscosity solutions. There are parabolic versions for the heat equation, where the maximum is attained on the bottom or sides of a space-time cylinder.