Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇², or ∆. The Laplacian ∆f(p) of a function f at a point p, up to a constant depending on the dimension, is the rate at which the average value of f over spheres centered at p, deviates from f(p) as the radius of the sphere grows. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems such as cylindrical and spherical coordinates, the Laplacian also has a useful form.

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, where the operator gives a constant multiple of the mass density when it is applied to a given gravitational potential. Solutions of the equation ∆f = 0, now called Laplace's equation, are the so-called harmonic functions, and represent the possible gravitational fields in free space.

The Laplacian occurs in differential equations that describe many physical phenomena, such as electric and gravitational potentials, the diffusion equation for heat and fluid flow, wave propagation, and quantum mechanics. The Laplacian represents the flux density of the gradient flow of a function. For instance, the net rate at which a chemical dissolved in a fluid moves toward or away from some point is proportional to the Laplacian of the chemical concentration at that point; expressed symbolically, the resulting equation is the diffusion equation. For these reasons, it is extensively used in the sciences for modelling all kinds of physical phenomena. The Laplacian is the simplest elliptic operator, and is at the core of Hodge theory as well as the results of de Rham cohomology. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection.

Definition

The Laplace operator is a second order differential operator in the n-dimensional Euclidean space, defined as the divergence (∇·) of the gradient (∇ƒ). Thus if ƒ is a twice-differentiable real-valued function, then the Laplacian of ƒ is defined by

$\Delta f = \nabla^2 f = \nabla \cdot \nabla f$

(1)

where the latter notations derive from formally writing $\nabla = \left ( \frac{\partial}{\partial x_1} , \dots , \frac{\partial}{\partial x_n} \right ).$ Equivalently, the Laplacian of ƒ is the sum of all the unmixed second partial derivatives in the Cartesian coordinates $x_i$ :

$\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}$

(2)

As a second-order 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 ∆ : C^k(Rⁿ) → C^k−2(Rⁿ), or more generally an operator ∆ : C^k(Ω) → C^k−2(Ω) for any open set Ω.

Motivation

Diffusion

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:

$\int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0,$

where n is the outward unit normal to the boundary of V. By the divergence theorem,

$\int_V \operatorname{div} \nabla u\, dV = \int_{\partial V} \nabla u \cdot \mathbf{n}\, dS = 0.$

Since this holds for all smooth regions V, it can be shown that this implies

$\operatorname{div} \nabla u = \Delta u = 0.$

The left-hand side of this equation is the Laplace operator. 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.

Density associated to a potential

If φ denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is given by the negative of the Laplacian of φ:

$q = -\epsilon_0\Delta\varphi\,$

where $\epsilon_0$ is the electric constant.

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 proportional to the charge enclosed:

$\int_{\partial V} \mathbf{E}\cdot \mathbf{n}\, dS = \int_V \operatorname{div}\mathbf{E}\,dV=\frac1{\epsilon_0}\int_V q\,dV.$

where the first equality is due to the divergence theorem. Since the electrostatic field is the (negative) gradient of the potential, this now gives

$-\int_V \operatorname{div}(\operatorname{grad}\phi)\,dV = \frac1{\epsilon_0}\int_V q\,dV.$

So, since this holds for all regions V, we must have

$\operatorname{div}(\operatorname{grad}\phi) = -\frac{1}{\epsilon_0}q$

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.

Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions to $\Delta f = 0$ in a region U are functions that make the Dirichlet energy functional stationary:

$E(f) = \frac{1}{2} \int_U \Vert \nabla f \Vert^2 \,dx.$

To see this, suppose $f\colon U\to \mathbb{R}$ is a function, and $u\colon U\to \mathbb{R}$ is a function that vanishes on the boundary of U. Then

$\frac{d}{d\varepsilon}\Big|_{\varepsilon = 0} E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, dx = -\int_U u \Delta f\, dx$

where the last equality follows using Green's first identity. This calculation shows that if $\Delta f = 0$, then E is stationary around f. Conversely, if E is stationary around f, then $\Delta f=0$ by the fundamental lemma of calculus of variations.

Coordinate expressions

Two dimensions

The Laplace operator in two dimensions is given by

$\Delta f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}$

where x and y are the standard Cartesian coordinates of the xy-plane.

In polar coordinates,

$\begin{align} \Delta f &= {1 \over r} {\partial \over \partial r} \left( r {\partial f \over \partial r} \right) + {1 \over r^2} {\partial^2 f \over \partial \theta^2}\\ &= {\partial^2 f \over \partial r^2} +{1 \over r} {\partial f \over \partial r} + {1 \over r^2} {\partial^2 f \over \partial \theta^2} . \end{align}$

Three dimensions

In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.

In Cartesian coordinates,

$\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.$

In cylindrical coordinates,

$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 \over \rho^2} {\partial^2 f \over \partial \varphi^2} + {\partial^2 f \over \partial z^2 }.$

In spherical coordinates:

$\Delta f = {1 \over r^2} {\partial \over \partial r} \left(r^2 {\partial f \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left(\sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}.$

(here φ represents the azimuthal angle and θ the zenith angle or co-latitude).

In general curvilinear coordinates ($\xi^1, \xi^2, \xi^3$):

$\nabla^2 = \nabla \xi^m \cdot \nabla \xi^n {\partial^2 \over \partial \xi^m \partial \xi^n} + \nabla^2 \xi^m {\partial \over \partial \xi^m },$

where summation over the repeated indices is implied.

N dimensions

In spherical coordinates in N dimensions, with the parametrization x = r θ ∈ R^N with r representing a positive real radius and θ an element of the unit sphere S^N−1,

$\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{N-1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \Delta_{S^{N-1}} f$

where $\Delta_{S^{N-1}}$ is the Laplace–Beltrami operator on the (N−1)-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as

$\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \Bigl(r^{N-1} \frac{\partial f}{\partial r} \Bigr).$

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.

Spectral theory

The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction ƒ with

$-\Delta f = \lambda f.$

This is known as the Helmholtz equation. 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 Kondrachov 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 n-sphere, the eigenfunctions of the Laplacian are the well-known spherical harmonics.

Generalizations

Laplace–Beltrami operator

The Laplacian also can 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, when applied to a function, is the trace of the function's Hessian:

$\Delta f = \mathrm{tr}(H(f))\,\!$

where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields, by a similar formula.

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

$\Delta f = d^* d f\,$

Here d^∗ is the codifferential, which can also be expressed using the Hodge dual. Note that this operator differs in sign from the "analyst's Laplacian" defined above, a point which must always be kept in mind when reading papers in global analysis. More generally, the "Hodge" Laplacian is defined on differential forms α by

$\Delta \alpha = d^* d\alpha + dd^*\alpha.\,$

This is known as the Laplace–de Rham operator, which is related to the Laplace–Beltrami operator by the Weitzenböck identity.

D'Alembertian

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 Laplace–Beltrami operator becomes the d'Alembert operator or d'Alembertian:

$\square = \frac {1}{c^2}{\partial^2 \over \partial t^2 } - {\partial^2 \over \partial x^2 } - {\partial^2 \over \partial y^2 } - {\partial^2 \over \partial z^2 }.$

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. Note that 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.

See also

• The vector Laplacian operator, a generalization of the Laplacian to vector fields.
• The Laplacian in differential geometry.
• The discrete Laplace operator is a finite-difference analog of the continuous Laplacian, defined on graphs and grids.
• The Laplacian is a common operator in image processing and computer vision (see the Laplacian of Gaussian, blob detector, and scale space).
• The list of formulas in Riemannian geometry contains expressions for the Laplacian in terms of Christoffel symbols.
• Weyl's lemma (Laplace equation)
• Earnshaw's theorem which shows that stable static gravitational, electrostatic or magnetic suspension is impossible
• Other situations in which a laplacian is defined are: analysis on fractals, time scale calculus and discrete exterior calculus.

Notes

1. Evans 1998, §2.2
2. Gilbarg & Trudinger 2001, Theorem 8.6
3. Gilbarg & Trudinger 2001, Corollary 8.11

References

• Evans, L (1998), Partial Differential Equations, American Mathematical Society, ISBN 978-0-8218-0772-9.
• Feynman, R, Leighton, R, and Sands, M (1970), "Chapter 12: Electrostatic Analogs", The Feynman Lectures on Physics, Volume 2, Addison-Wesley-Longman.
• Gilbarg, D.; Trudinger, N. (2001), Elliptic partial differential equations of second order, Springer, ISBN 978-3-540-41160-4.
• Schey, H. M. (1996), Div, grad, curl, and all that, W W Norton & Company, ISBN 978-0-393-96997-9.

External links

• Hazewinkel, Michiel, ed. (2001), "Laplace operator", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W., "Laplacian", MathWorld.