Here is a genuinely powerful, and genuinely intuitive, fact about steady-state temperature, steady electric potential, or any other harmonic quantity: it can never have a local maximum or minimum sitting strictly inside its domain. The hottest and coldest spots of a metal plate that has settled into equilibrium always occur somewhere on the plate's edge — never buried in the middle. That sounds almost too clean to be a real mathematical theorem, but it has real teeth, and we can prove it outright.

Think about why this has to be physically sensible first. If some interior point really were strictly hotter than every point around it, heat would immediately start flowing away from it in all directions — down the temperature gradient — and the temperature there would drop. A genuine steady state, one that isn't changing anymore, simply can't sustain a hidden interior hot spot. The maximum principle turns that physical intuition into an ironclad mathematical guarantee.

Why: it follows from the mean-value property

The proof idea is short and comes directly from solving Laplace's equation on a disk, where we found that the value of a harmonic function at the centre of any disk equals the average of its values around that disk's boundary circle.

Now suppose, for contradiction, that some interior point p were a strict local maximum — strictly bigger than every one of its neighbours. Draw a tiny circle around p, entirely inside the domain. By the mean-value property, u(p) must equal the average of u on that little circle. But an average can never be strictly bigger than every single value being averaged — it has to sit somewhere between the smallest and the largest of them, and it can only equal the largest if every value on the circle is exactly that same largest value. So u(p) being a strict maximum is impossible unless u is constant nearby. No interior point can be a lone, strict peak.

Worked example: a plate heated unevenly around its edge

Take a circular metal plate whose rim is welded to a ring of thermostats reading anywhere from 20^\circ\text{C} up to 80^\circ\text{C} at different points around the edge. Once the plate settles into a steady state, what's the hottest possible temperature anywhere inside it?

The maximum principle answers this immediately, without solving a single equation: the interior can be at most as hot as the hottest edge reading — 80^\circ\text{C} — and never hotter. Likewise, nowhere inside can it dip below the coldest edge reading of 20^\circ\text{C}. Every interior value is squeezed between the boundary's coldest and hottest points.

The picture below shows a simpler, fully solvable case: rim data f(\theta) = \cos\theta, so (as we found on the disk page) u(r,\theta) = r\cos\theta = x. Its extreme values, +1 and -1, sit exactly at the rightmost and leftmost points of the rim — nowhere in the interior does the colour get darker than it is right at the edge, and the very centre lands exactly on the boundary average, 0.

The same guarantee works on any shape, not just circles. Picture a square metal plate whose bottom edge is heated to a steady 100^\circ\text{C} while the other three edges are held at 0^\circ\text{C} — the classic rectangle example. Without solving the Fourier series at all, the maximum principle already tells us the interior temperature can never reach, let alone exceed, 100^\circ\text{C}: that's the largest value appearing anywhere on the boundary, so it's an absolute ceiling for every point inside. It also can't dip below 0^\circ\text{C}. The full series solution will, of course, confirm this — but the maximum principle lets us know the answer's range before we've done a scrap of the actual work.

This is also a genuine, practical sanity-check for numerical simulations. If you solve a steady-state heat problem on a computer and the output shows an interior grid cell reading hotter than every boundary cell, that is not a legitimate physical result — it is a red flag that a bug exists somewhere in the code (a sign error, a bad mesh cell, a solver that hasn't converged). The maximum principle is a free, automatic check that costs nothing to run.

Why it guarantees uniqueness

The maximum principle is not just a curiosity — it is the cleanest route to uniqueness for the Dirichlet problem. Suppose two harmonic functions u_1 and u_2 share the same boundary values. Their difference w = u_1 - u_2 is harmonic and is 0 on the entire boundary. By the maximum principle its maximum and minimum are both 0, so w \equiv 0 throughout — meaning u_1 = u_2. The boundary data determines the solution completely.

The same argument gives stability: if two boundary datasets differ by at most \varepsilon, their solutions differ by at most \varepsilon everywhere inside. Existence, uniqueness, and continuous dependence — the Dirichlet problem for Laplace's equation is well-posed.

This is exactly why the sanity check above works: two slightly different numerical runs of the same physical setup, with boundary data agreeing to within a tiny tolerance, are guaranteed to agree everywhere inside to within that same tolerance. A simulation that drifts far from a known solution despite matching boundary conditions closely is, again, telling you something is wrong.