This equation governs the velocity potential of an incompressible flow and the steady-state distribution of heat.
Solutions of this equation that map to Euclidean space correspond to instantons with action given by the Dirichlet energy. For a certain set of solutions to this equation, taking the complex conjugate is equivalent to negating the second degree parameter and multiplying by the Condon-Shortley phase. Those solutions to this equation, the simplest of which is one-half times the square root of one over pi, are eigenfunctions of the square of the orbital angular momentum operator. Substituting this equation into the expression for the energy of a dipole in a constant magnetic field results in Earnshaw's theorem. In two dimensions, this equation's fundamental solution is "negative log r over two pi," and in three dimensions, it is the electric potential generated by a point charge. This equation governs the velocity potential of an incompressible flow and the steady-state distribution of heat. For 10 points, harmonic functions satisfy what equation written "del squared u equals zero"?
Solutions of this equation that map to Euclidean space correspond to instantons with action given by the Dirichlet energy.
Laplace's equation [do not accept or prompt on more general answers such as "Poisson's equation" or "heat equation"]