Laplace's equation

Summary

Laplace's equation is the partial differential equation $\Delta u = 0$, where $\Delta$ is the Laplacian, the sum of the second partial derivatives. Its solutions are the harmonic functions. It is the model elliptic equation. It describes steady states: equilibrium temperatures, electrostatic and gravitational potentials in empty space, and potential flows of an ideal fluid. The Dirichlet problem asks for a solution with given values on the boundary of a region.