Comparison principle

Remarks

Proof idea: the function $w = u - v$ satisfies $L w \ge 0$ in $\Omega$ and $w \le 0$ on $\partial\Omega$. The weak maximum principle for operators with $c \le 0$ gives $\sup_\Omega w \le \sup_{\partial\Omega} w^+ = 0$.

Example: if $\Delta u \ge 0$ in the unit ball $B$ and $u \le 1$ on $\partial B$, comparing with $v = 1$ gives $u \le 1$ in $B$. If $-\Delta u = 1$ in $B \subseteq \R^n$ with $u = 0$ on $\partial B$, comparison with $0$ and with $(1 - \abs{x}^2)/(2n)$ shows $0 \le u \le (1 - \abs{x}^2)/(2n)$; in fact $u$ equals this function.

The sign condition on $c$ is needed: for $L u = u'' + u$ on $(0, \pi)$, the functions $u = \sin x$ and $v = 0$ satisfy $L u = L v = 0$ and $u = v$ at the endpoints, but $u > v$ inside.

Uniqueness for the Dirichlet problem $L u = f$ in $\Omega$, $u = g$ on $\partial\Omega$, follows by applying the theorem twice.

Going deeper

For a student

The comparison principle is the main use of the maximum principle. It gives uniqueness for boundary value problems, a priori bounds by explicit barriers, and monotone iteration schemes between a subsolution and a supersolution. Versions exist for parabolic equations, through the parabolic maximum principle, and for fully nonlinear equations, where comparison for viscosity solutions is the key uniqueness tool. It is the PDE analogue of the comparison theorem for ordinary differential equations.