8 Laplace’s Equation

When the temperature in a body stops changing, \(u_t = 0\) and the heat equation becomes Laplace’s equation \[ \Delta u = 0 . \] Its solutions, harmonic functions, are also the electrostatic potential in a region free of charge, the gravitational potential in empty space, the velocity potential of an incompressible irrotational flow, and the real and imaginary parts of analytic functions of a complex variable. With a source, \(-\Delta u = f\) is Poisson’s equation. Harmonic functions are extraordinarily rigid: their value at a point is the average of their values around it, they have no interior maxima, and their boundary values determine them completely. This section proves these facts and solves the problem in a disc.

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.