Differential equations MOC

Laplace's equation

Laplace's differential equation is the statement that a scalar-valued function's Laplacian is zero.

Properties of solutions

A solution to Laplace's equation has the following properties in general

• It has the shape of something under tension, e.g. a rubber band or rubber sheet under tension.
• Let , and be the set of all points with distance from . Then the average of the image is .
• The value of within a region is completely and uniquely determined by the values of on the boundary .
• There are no local maxima or minima (this follows from the previous property)

Holomorphic extension

Any function satisfying Laplace's equation may be extended to a holomorphic function with unique up to an imaginary constant. This follows directly from the Cauchy-Riemann equations.

See also

• Poisson's equation

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