---
tag: D2JF19HF91
kind: statement
role: theorem
name: Maximum principle
msc: [35B50, 35J15]
uses: [XY8GEQSWPR, S8RG6N1PRT, 0VW90BBSJ4]
about: [XY8GEQSWPR]
sources:
  - url: https://en.wikipedia.org/wiki/Maximum_principle
---

## Statement

Let $U \subset \R^n$ be a bounded open set.
Let
$$L u = \sum_{i,j=1}^{n} a_{ij}(x) \, \partial_i \partial_j u + \sum_{i=1}^{n} b_i(x) \, \partial_i u$$
with continuous coefficients on the closure $\overline{U}$, where $a_{ij} = a_{ji}$.
Assume that $L$ is uniformly [[elliptic|XY8GEQSWPR]]: there is $\theta > 0$ with $\sum_{i,j} a_{ij}(x) \xi_i \xi_j \ge \theta \abs{\xi}^2$ for all $x \in U$ and $\xi \in \R^n$.
Let $u$ be twice continuously differentiable on $U$ and continuous on $\overline{U}$, with $L u \ge 0$ in $U$.

1. (Weak maximum principle.) $\max_{\overline{U}} u = \max_{\partial U} u$.
2. (Strong maximum principle.) If moreover $U$ is connected and $u$ attains its maximum over $\overline{U}$ at a point of $U$, then $u$ is constant on $U$.

## 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|S8RG6N1PRT]] 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 $u^+ = \max(u, 0)$.

## Explanations

### L1 human

The maximum principle says that a solution of an elliptic equation cannot have a peak inside the region: its largest value is on the boundary.
For a temperature in equilibrium, the hottest point is on the edge, since a hot spot inside would spread its heat to the cooler surroundings.
If the largest value is reached inside a connected region, the solution must be constant.

### L2 human

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.