8.2 The mean value property
Theorem 8.2.1 (Mean value property). If \(u\) is harmonic in an open set \(\Omega \) and the closed ball \(\overline {B_r(x)} \subset \Omega \), then \(u(x)\) equals the average of \(u\) over the sphere \(\partial B_r(x)\) and over the ball \(B_r(x)\).
Proof. Let \(\varphi (\rho ) = M_u(x, \rho )\), the spherical mean of Section 5. The calculation in the proof of Darboux’s lemma gives \(\varphi '(\rho ) = \frac {1}{\omega _n\rho ^{n - 1}}\int _{B_\rho (x)}\Delta u\,dy = 0\), so \(\varphi \) is constant on \((0, r]\), and \(\varphi (\rho ) \to u(x)\) as \(\rho \to 0\) by continuity. The ball average is an average of sphere averages, each equal to \(u(x)\). □
The converse holds too: a continuous function with the mean value property on every small sphere is harmonic, and in particular \(C^\infty \). The property is the analogue of the five-point rule \(u_{i,j} = \frac 14(u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1})\) for the discrete Laplace equation, and it is the reason Laplace’s equation describes equilibria: each point is in balance with its neighbours.
Theorem 8.2.2 (Maximum principle). Let \(u\) be harmonic in a connected open set \(\Omega \).
- (i)
- Strong form. If \(u\) attains its maximum over \(\Omega \) at a point of \(\Omega \), then \(u\) is constant.
- (ii)
- Weak form. If \(\Omega \) is bounded and \(u\) is continuous on \(\overline \Omega \), then \(\max _{\overline \Omega }u = \max _{\partial \Omega }u\) and \(\min _{\overline \Omega }u = \min _{\partial \Omega }u\).
Proof. (i) Let \(M = \max u\) and \(S = \{x \in \Omega : u(x) = M\}\), non-empty by hypothesis and closed in \(\Omega \) by continuity. If \(x \in S\) and \(B_r(x) \subset \Omega \), then \(M = u(x)\) is the average of \(u \leq M\) over the ball, so \(u = M\) on the ball: \(S\) is open. Since \(\Omega \) is connected, \(S = \Omega \). (ii) \(u\) attains its maximum over the compact set \(\overline \Omega \). If it does so only at interior points, (i) applied to the component containing one of them makes \(u\) constant there, and by continuity the same value is attained on the boundary of that component, which is part of \(\partial \Omega \). For the minimum apply this to \(-u\). □
The consequences are those for the heat equation. The Dirichlet problem \[ -\Delta u = f \text { in } \Omega , \qquad u = g \text { on } \partial \Omega , \] has at most one solution continuous on \(\overline \Omega \), and solutions depend continuously on the boundary data: \(\max |u_1 - u_2| \leq \max _{\partial \Omega }|g_1 - g_2|\). If \(g \geq 0\) then \(u \geq 0\), and if \(g\) is not identically \(0\), then \(u > 0\) throughout. For the Neumann problem, \(\partial _\nu u = g\) on the boundary, Green’s identity shows that solutions differ by constants (the difference \(w\) has \(\int |\nabla w|^2 = \int _{\partial \Omega }w\,\partial _\nu w = 0\)) and that a solution can exist only if \(\int _{\partial \Omega }g\,dS = -\int _\Omega f\,dx\): in a steady state the heat flowing in through the boundary must balance the heat produced inside.
A physical consequence is Earnshaw’s theorem (1842). In a region free of charge the electrostatic potential is harmonic, so by the strong maximum principle it has no strict local minimum; a charged particle in such a field has no point of stable equilibrium. Charges cannot be held in place by static electric fields alone, which is why ion traps use oscillating fields.
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