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Laplace operator
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In mathematics and physics, the Laplace operator or Laplacian, denoted by or and named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. In physics, it is used in modeling of wave propagation, heat flow and forming the Helmholtz equation. It is central in electrostatics and fluid mechanics, anchoring in Laplace's equation and Poisson's equation. In quantum mechanics, it represents the kinetic energy term of the Schrödinger equation. In mathematics, functions with vanishing Laplacian are called harmonic functions; the Laplacian is at the core of Hodge theory and the results of de Rham cohomology.
Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence of the gradient .
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Encyclopedia
In mathematics and physics, the Laplace operator or Laplacian, denoted by or and named after Pierre-Simon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. In physics, it is used in modeling of wave propagation, heat flow and forming the Helmholtz equation. It is central in electrostatics and fluid mechanics, anchoring in Laplace's equation and Poisson's equation. In quantum mechanics, it represents the kinetic energy term of the Schrödinger equation. In mathematics, functions with vanishing Laplacian are called harmonic functions; the Laplacian is at the core of Hodge theory and the results of de Rham cohomology.
Definition
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 is a twice-differentiable real-valued function, then the Laplacian of f is defined by
(1)
Equivalently, the Laplacian of f is the sum of all the unmixed second partial derivatives in the Cartesian coordinates :
(2)
As a second-order differential operator, the Laplace operator maps Ck-functions to Ck-2-functions for k = 2. The expression (1) (or equivalently (2)) defines an operator ? : Ck(Rn) ? Ck-2(Rn), or more generally an operator ? : Ck(O) ? Ck-2(O) for any open set O.
The Laplacian of a function is also the trace of the function's Hessian, a definition more accessible from linear algebra and statistics:
Motivation
Diffusion
In the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematical description of equilibrium. Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary, of any smooth region V is zero, provided there is no source or sink within V:
where n is the unit normal to the boundary of V. By the divergence theorem,
Since this holds for all smooth regions V, it can be shown that this implies
-
The left-hand side of this equation is the Laplace operator.
Energy minimization
Another motivation for the Laplacian appearing in physics is that solutions to in a region U are functions that make the energy functional stationary.
-
To see this, suppose
is a function, and
is a function that vanishes on the
boundary of U. Then
-
where the last equality follows using Green's first identity.
This calculation shows that if , then
E is stationary around f. Conversely, if E is stationary
around f, then by the fundamental lemma of calculus of variations.
Coordinate expressions
Two dimensions
The Laplace operator in two dimensions is given by
where x and y are the standard Cartesian coordinates of the xy-plane.
In polar coordinates,
Three dimensions
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
In Cartesian coordinates,
In cylindrical coordinates,
In spherical coordinates:
(here represents the polar angle and the azimuthal angle). The term can be replaced by its equivalent as well. See also the article Del in cylindrical and spherical coordinates.
N dimensions
In spherical coordinates in dimensions, with the parametrization with and ,
where is the Laplace–Beltrami operator on the dimensional sphere, or spherical Laplacian. One can also write the term equivalently as .
As a consequence, the spherical Laplacian of a function defined on can be computed as the ordinary Laplacian of the function extended to so that it is constant along rays.
Identities
- If f and g are functions, then the Laplacian of the product is given by
Note the special case where f is a radial function and g is a spherical harmonic, . One encounters this special case in numerous physical models. The gradient of is a radial vector and the gradient of an angular function is tangent to the radial vector, therefore
In addition, the spherical harmonics have the special property of being eigenfunctions of the angular part of the Laplacian in spherical coordinates.
Therefore,
Spectral theory
The spectrum of the Laplace operator consists of all eigenvalues ? for which there is a corresponding eigenfunction ƒ with
If O is a bounded domain in Rn then the eigenfunctions of the Laplacian are an orthonormal basis in the Hilbert space L2(Ω). This result essentially follows from the spectral theorem on compact self-adjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and Kondrakov embedding theorem). It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary. When O is the n-sphere, the eigenfunctions of the Laplacian are the well-known spherical harmonics.
Generalizations
The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic.
In the Minkowski space the Laplacian becomes the d'Alembert operator or d'Alembertian:
The D'Alembert operator is often used to express the Klein–Gordon equation and the four-dimensional wave equation. The signs in front of the spatial derivatives are negative, while they would have been positive in the Euclidean space. The additional factor of c is needed if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, physicists usually work in units such that c=1 in order to simplify the equation.
Laplace–Beltrami operator
The Laplacian can also be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds. The Laplace–Beltrami operator can also be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields.
Another way to generalize the Laplace operator to pseudo-Riemannian manifolds is via the Laplace – de Rham operator which operates on differential forms. This is then related to the Laplace–Beltrami operator by the Weitzenböck identity.
See also
Footnotes
External links
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