Maximum principles and uniqueness

Weak maximum principle. Let \( \Omega \) be a bounded domain and \( u \) be continuous on its closure and twice differentiable inside, with \( \Delta u \ge 0 \) (subharmonic). Then \( \max_{\overline\Omega}u = \max_{\partial\Omega}u \). First suppose \( \Delta v > 0 \). At an interior maximum each pure second derivative is \( \le 0 \), so \( \Delta v \le 0 \), a contradiction; the maximum must be on the boundary. For \( \Delta u \ge 0 \) set \( v = u + \varepsilon|x|^2 \), so \( \Delta v = \Delta u + 2n\varepsilon > 0 \). If \( \Omega \) lies in a ball of radius \( R \) around the origin, \( u \le v \le \max_{\partial\Omega}v \le \max_{\partial\Omega}u + \varepsilon R^2 \), and letting \( \varepsilon \to 0 \) finishes it.

Strong maximum principle. If a harmonic function on a connected domain attains its maximum at an interior point, it is constant. By the mean value property, the value at a maximum point is an average of values that are all no larger, so they all equal it; the set where the maximum is attained is therefore open, it is closed by continuity, and connectedness makes it everything.

Uniqueness and stability. If \( u_1, u_2 \) solve \( \Delta u = f \) in \( \Omega \) with boundary values \( g_1, g_2 \), then \( w = u_1 - u_2 \) is harmonic, so \( \max_{\overline\Omega}|w| \le \max_{\partial\Omega}|g_1 - g_2| \). Equal boundary data give equal solutions, and close data give close solutions, uniformly. For the heat equation on a space-time cylinder the maximum is attained on the parabolic boundary: the initial time slice together with the sides. So a rod whose initial temperature is \( x(1 - x) \) and whose ends are held at zero never exceeds the value \( 1/4 \) that the example finds.

Picture it: no hot spot can appear in the middle of a room out of nothing; the hottest point is always at a heater (the boundary) or was already hot at the start.

Think it: maximum principles are about signs, and they belong to scalar equations. They give comparison (a subsolution stays below a supersolution) and bounds in the uniform norm with no integration. They fail for systems: there is no maximum principle for the size of the Navier-Stokes velocity in three dimensions, which is one reason the problem is hard. In two dimensions the vorticity is a scalar that is transported and diffused, it does satisfy one, and that is part of why two-dimensional flows are globally regular.