Determining solutions to this over a region based on boundary conditions was the original formulation of Dirichlet's problem, and the lack of local maxima or minima in its solutions provides an easy proof of Earnshaw's Theorem.
Determining solutions to this over a region based on boundary conditions was the original formulation of Dirichlet's problem, and the lack of local maxima or minima in its solutions provides an easy proof of Earnshaw's Theorem. Solutions can be found numerically using the method of relaxation, which takes advantage of the mean value property satisfied by its solutions. In electrostatics it describes the potential in a region with no charge, and it is the homogenous form of Poisson's equation. FTP, name this partial differential equation with solutions known as harmonic functions in which the divergence of the gradient of a scalar function vanishes, named for a French mathematician.
Determining solutions to this over a region based on boundary conditions was the original formulation of Dirichlet's problem, and the lack of local maxima or minima in its solutions provides an easy proof of Earnshaw's Theorem.
{Laplace}'s Equation