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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 operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient (). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by:
The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation Δ ∞ u = 0 {\displaystyle \Delta _{\infty }u=0} , the two definitions are equivalent.
In the study of hyperbolic and parabolic partial differential equations, zeros of the principal symbol correspond to the characteristics of the partial differential equation. In applications to the physical sciences, operators such as the Laplace operator play a major role in setting up and solving partial differential equations.
In general solutions of equations involving the p-Laplacian do not have second order derivatives in classical sense, thus solutions to these equations have to be understood as weak solutions. For example, we say that a function u belonging to the Sobolev space W 1 , p ( Ω ) {\displaystyle W^{1,p}(\Omega )} is a weak solution of
For any twice-differentiable real-valued function f defined on Euclidean space R n, the Laplace operator (also known as the Laplacian) takes f to the divergence of its gradient vector field, which is the sum of the n pure second derivatives of f with respect to each vector of an orthonormal basis for R n.
A solution to Laplace's equation defined on an annulus.The Laplace operator is the most famous example of an elliptic operator.. In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator.
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