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In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
The Laplace pressure is the pressure difference between the inside and the outside of a curved surface that forms the boundary between two fluid regions. [1] The pressure difference is caused by the surface tension of the interface between liquid and gas, or between two immiscible liquids.
Diagram illustrating the image method for Laplace's equation for a sphere of radius R. The green point is a charge q lying inside the sphere at a distance p from the origin, the red point is the image of that point, having charge −qR/p, lying outside the sphere at a distance of R 2 /p from the origin. The potential produced by the two charges ...
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation ...
Laplace's equation in spherical coordinates, such as are used for mapping the sky, can be simplified, using the method of separation of variables into a radial part, depending solely on distance from the centre point, and an angular or spherical part. The solution to the spherical part of the equation can be expressed as a series of Laplace's ...
A function (or, more generally, a distribution) is weakly harmonic if it satisfies Laplace's equation = in a weak sense (or, equivalently, in the sense of distributions). A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth.
In this equation, we used sup and inf instead of max and min because the graph (,) does not have to be locally finite (i.e., to have finite degrees): a key example is when () is the set of points in a domain in , and (,) if their Euclidean distance is at most . The importance of this example lies in the following.
Relaxation methods are used to solve the linear equations resulting from a discretization of the differential equation, for example by finite differences. [ 2 ] [ 3 ] [ 4 ] Iterative relaxation of solutions is commonly dubbed smoothing because with certain equations, such as Laplace's equation , it resembles repeated application of a local ...