The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δf = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum.
The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials; the diffusion equation describes heat and fluid flow, the wave equation describes wave propagation, and the Schrödinger equation in quantum mechanics. In image processing and computer vision, the Laplacian operator has been used for various tasks, such as blob and edge detection. The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.
As a second-order differential operator, the Laplace operator maps Ck functions to Ck−2 functions for k ≥ 2. It is a linear operator Δ : Ck(Rn) → Ck−2(Rn), or more generally, an operator Δ : Ck(Ω) → Ck−2(Ω) for any open set Ω ⊆ Rn.
In the physical theory of diffusion, the Laplace operator 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 ∂V of any smooth region V is zero, provided there is no source or sink within V:
Since this holds for all smooth regions V, one can show that it implies:
The Laplace operator itself has a physical interpretation for non-equilibrium 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. This interpretation of the Laplacian is also explained by the following fact about averages.
This is a consequence of Gauss's law. Indeed, if V is any smooth region with boundary ∂V, then by Gauss's law the flux of the electrostatic field E across the boundary is proportional to the charge enclosed:
The same approach implies that the negative of 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.
To see this, suppose f : U → R is a function, and u : U → R 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.
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.
If Ω is a bounded domain in Rn, then the eigenfunctions of the Laplacian are an orthonormal basis for 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 the Rellich–Kondrachov 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, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Ω is the n-sphere, the eigenfunctions of the Laplacian are the spherical harmonics.
For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.
And, in the same manner, a dot product, which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices:
Another example is the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents:
A version of the Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms. For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.
The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold. The Laplace–Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian:
Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative, in terms of which the "geometer's Laplacian" is expressed as
Here δ is the codifferential, which can also be expressed in terms of the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on differential forms α by
It is the generalisation of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy particle physics. The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the wave equations, and it is also part of the Klein–Gordon equation, which reduces to the wave equation in the massless case.
The additional factor of c in the metric is needed in physics 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, theoretical physicists usually work in units such that c = 1 in order to simplify the equation.
The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.