6th  Top partial differential equation topics 
In mathematics and physics, the Laplace operator or Laplacian, named after PierreSimon de Laplace, is a differential operator, specifically an important case of an elliptic operator, with many applications. It is denoted by the symbols Δ, ∇^{2}, or ∇·∇. In physics, it is used in the modeling of wave propagation, heat flow, and fluid mechanics. It is central in electrostatics, where it represents the charge associated to a given potential. It is the main part of Laplace's equation for a conservative potential, Poisson's equation for the gravitational potential associated to a given mass, and the Helmholtz equation for the vibrations of a drum. 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. In image processing and computer vision, the Laplacian is used as a blob detector for tasks such as object recognition and texture analysis as well as in early works on edge detection.
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The Laplace operator is a second order differential operator in the ndimensional Euclidean space, defined as the divergence () of the gradient (). Thus if f is a twicedifferentiable realvalued function, then the Laplacian of f is defined by
Equivalently, the Laplacian of f is the sum of all the unmixed second partial derivatives in the Cartesian coordinates x_{i}:
As a secondorder differential operator, the Laplace operator maps C^{k}functions to C^{k−2}functions for k ≥ 2. The expression (1) (or equivalently (2)) defines an operator Δ : C^{k}(R^{n}) → C^{k−2}(R^{n}), or more generally an operator Δ : C^{k}(Ω) → C^{k−2}(Ω) for any open set Ω.
The Laplacian of a function is also the trace of the function's Hessian:
In the physical theory of diffusion, the Laplace operator (via Laplace's equation) arises naturally in the mathematical description of equilibrium.^{[1]} 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 outward 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 lefthand side of this equation is the Laplace operator. The Laplace operator itself has a physical interpretation for nonequilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation.
If φ denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is given by the Laplacian of φ:

( 
This is a consequence of Gauss's law. Indeed, if V is any smooth region, then by Gauss's law the flux of the electrostatic field E is equal to the charge enclosed (in appropriate units):
where the first equality uses the fact that the electrostatic field is the gradient of the electrostatic potential. The divergence theorem now gives
and since this holds for all regions V, (1) follows.
The same approach implies that the Laplacian of the gravitational potential is the mass distribution. Often the charge (or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation.
Another motivation for the Laplacian appearing in physics is that solutions to Δf = 0 in a region U are functions that make the Dirichlet 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 Δf = 0, then E is stationary around f. Conversely, if E is stationary around f, then Δf = 0 by the fundamental lemma of calculus of variations.
The Laplace operator in two dimensions is given by
where x and y are the standard Cartesian coordinates of the xyplane.
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
(here φ represents the azimuthal angle and θ the polar angle).
In spherical coordinates in N dimensions, with the parametrization x = rθ ∈ R^{ N} with r representing a positive real radius and θ an element of the unit sphere S^{N−1},
where is the Laplace–Beltrami operator on the (N−1)sphere, known as the spherical Laplacian. The two radial terms can be equivalently rewritten as
As a consequence, the spherical Laplacian of a function defined on S^{N−1} ⊂ R^{ N} can be computed as the ordinary Laplacian of the function extended to R^{N}\{0} so that it is constant along rays, i.e., homogeneous of degree zero.
The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction ƒ with
If Ω is a bounded domain in R^{n} then the eigenfunctions of the Laplacian are an orthonormal basis in the Hilbert space L^{2}(Ω). This result essentially follows from the spectral theorem on compact selfadjoint operators, applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and Kondrakov embedding theorem).^{[2]} It can also be shown that the eigenfunctions are infinitely differentiable functions.^{[3]} More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Ω is the nsphere, the eigenfunctions of the Laplacian are the wellknown spherical harmonics.
The Laplacian can be generalized in certain ways to nonEuclidean 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 also known as the wave operator, because it is the differential operator appearing in the fourdimensional wave equation. It is also the leading part of the Klein–Gordon 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.
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 pseudoRiemannian 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 pseudoRiemannian 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.
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Laplace operator (plural Laplace operators)
