Laplace's Equation and the Maximum Principle
A solution of Laplace's equation equals its own average over any small sphere, so it can have no interior peak. Uniqueness and stability for the boundary value problem follow from that one observation.
What a learner can do afterwards
- State the mean value property and derive the maximum principle from it
- Use the maximum principle to prove uniqueness for the Dirichlet problem
- Explain why harmonic functions are smooth even when the boundary data is rough
1 · Read
A harmonic function solves Laplace's equation u_xx plus u_yy equals 0. Every such function has the mean value property: its average over any circle equals its value at the center. For example, u(x,y) equals 3x plus 4 is harmonic, and its average over a circle centered at (1,1) is just u(1,1), which is 7. Not every formula qualifies: x squared plus y squared has Laplacian 4, while x squared minus y squared has Laplacian 0.
Take u(x,y) equals x squared minus y squared and a small circle centered at (2,1). Since u_xx is 2 and u_yy is negative 2, the function is harmonic. The mean value property says the circle average equals the center value u(2,1), which is 4 minus 1, giving 3. A linear function like 5x minus 12y plus 8 works the same way: its average over a circle at the origin is u(0,0), which is 8.
From averaging comes the maximum principle: a nonconstant harmonic function attains its maximum and minimum only on the boundary, never strictly inside. Uniqueness for the Dirichlet problem follows at once. If two solutions share the same boundary data, their difference is harmonic with zero boundary values, so both its max and min are zero and the solutions agree everywhere. Boundary values between 0 and 5 likewise trap the whole interior between 0 and 5.
Harmonic functions are smooth at every interior point even when the boundary data is rough and merely continuous. Averaging is what smooths: each interior value blends the whole boundary, so no interior point inherits a jump. Even a sharp spike in the boundary data leaves the interior finite and smooth. Think of electrostatics: with no interior charge, the boundary voltages fix one gentle landscape inside.
Averages equal center values, so peaks live on the boundary and boundary data fixes the interior.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.