the moving-plane method

Suppose a problem is perfectly round — a ball-shaped domain, with data that does not care about direction. You would expect the solution to be round too, depending only on the distance from the centre. But expecting is not proving. The moving-plane method is a beautiful technique that PROVES such solutions are radially symmetric, using nothing but the maximum principle and a clever reflection. It is the standard tool for turning the symmetry of a problem into the symmetry of its solution.

Here is the idea in plain steps. Pick a direction and a plane far on one side of the domain. Reflect the part of the solution on that side across the plane. Now slide the plane inward and, at each position, compare the solution with its reflection on the overlapping region. The maximum principle and the Hopf boundary-point lemma let you show that as long as the plane has not passed the centre, the reflected solution stays below (or equal to) the original — they never cross. Push the plane until it reaches the centre. By the same argument coming from the opposite side, the solution must equal its own reflection across the central plane. Since the direction was arbitrary, the solution is symmetric in every direction at once — that is, radially symmetric. A bonus from Hopf: the solution is also strictly monotone in the radial direction, decreasing as you move outward.

This was made famous by the Gidas-Ni-Nirenberg theorem, which shows positive solutions of many elliptic equations on a ball are automatically radial and radially decreasing, and by Serrin's overdetermined-problem result. It is one of the deepest payoffs of maximum-principle thinking: a purely qualitative method delivers exact, rigid geometric information about solutions you can rarely write down. Honesty note: it needs the maximum principle to apply (elliptic structure, sign conditions on the nonlinearity) and enough symmetry in the domain and data — it is powerful but not automatic.