Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , <math>\nabla^2</math> (where <math>\nabla</math> is the nabla operator), or . 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. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from .

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 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 describes the wave function 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. It is also essentially the infinitesimal generator of standard Brownian motion on .

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 <math>f</math> is a twice-differentiable real-valued function, then the Laplacian of <math>f</math> is the real-valued function defined by:

where the latter notations derive from formally writing:

<math display="block">\nabla = \left ( \frac{\partial }{\partial x_1} , \ldots , \frac{\partial }{\partial x_n} \right ).</math>

Explicitly, the Laplacian of is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates :

As a second-order differential operator, the Laplace operator maps functions to functions for . It is a linear operator , or more generally, an operator for any open set .

Alternatively, the Laplace operator can be defined as:

<math display="block">\nabla^2 f(\vec{x}) = \lim_{R \rightarrow 0} \frac{2n}{R^2} (f_{\text{shell}_R} - f(\vec{x})) = \lim_{R \rightarrow 0} \frac{2n}{A_{n-1} R^{1+n \int_{\text{shell}_R} f(\vec{r}) - f(\vec{x}) d r^{n-1} </math>

where <math>n</math> is the dimension of the space, <math>f_{\text{shell}_R}</math> is the average value of <math>f</math> on the surface of an n-sphere of radius , <math>\textstyle \int_{\text{shell}_R} f(\vec{r}) d r^{n-1}</math> is the surface integral over an -sphere of radius , and <math>A_{n-1}</math> is the hypervolume of the boundary of a unit -sphere.

Sign conventions

There is no single standard sign convention for the Laplace operator. In Euclidean coordinates, one common convention is

<math display="block">\Delta=\nabla\cdot\nabla=\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2},</math>

so that for every smooth compactly supported function <math>\varphi</math>,

<math display="block">\int_{\mathbf R^n}\overline{\varphi(x)}\,\Delta\varphi(x)\,dx

-\int_{\mathbf R^n} |\nabla \varphi(x)|^2\,dx,</math>

and hence <math>\Delta</math> is negative semidefinite on <math>L^2</math>.

Another common convention inserts a minus sign and defines instead

<math display="block">\Delta=-\nabla\cdot\nabla=-\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2},</math>

so that the Laplacian is nonnegative.

Both conventions occur in the literature, and authors usually state explicitly which one they are using. In this article, unless otherwise noted, <math>\Delta</math> denotes the Euclidean Laplacian

<math display="block">\Delta=\nabla\cdot\nabla=\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2}.</math>

Motivation

Diffusion

In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium. Specifically, if is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of through the boundary (also called ) of any smooth region is zero, provided there is no source or sink within :

<math display="block">\int_{S} \nabla u \cdot \mathbf{n}\, dS = 0,</math>

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

<math display="block">\int_V \operatorname{div} \nabla u\, dV = \int_{S} \nabla u \cdot \mathbf{n}\, dS = 0.</math>

Since this holds for all smooth regions , one can show that it implies:

<math display="block">\operatorname{div} \nabla u = \Delta u = 0.</math>

The left-hand side of this equation is the Laplace operator, and the entire equation is known as Laplace's equation. Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.

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.

Averages

Given a twice continuously differentiable function <math>f : \R^n \to \R </math> and a point , the average value of <math>f </math> over the ball with radius <math>h</math> centered at <math>p</math> is:

<math display="block">\overline{f}_B(p,h)=f(p)+\frac{\Delta f(p)}{2(n+2)} h^2 +o(h^2) \quad\text{for}\;\; h\to 0</math>

Similarly, the average value of <math>f </math> over the sphere (the boundary of a ball) with radius <math>h</math> centered at <math>p</math> is:

<math display="block">\overline{f}_S(p,h)=f(p)+\frac{\Delta f(p)}{2n} h^2 +o(h^2) \quad\text{for}\;\; h\to 0.</math>

Density associated with a potential

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

<math display="block">q = -\varepsilon_0 \Delta\varphi,</math>

where is the electric constant.

This is a consequence of Gauss's law. Indeed, if is any smooth region with boundary , then by Gauss's law the flux of the electrostatic field across the boundary is proportional to the charge enclosed:

<math display="block">\int_{\partial V} \mathbf{E}\cdot \mathbf{n}\, dS = \int_V \operatorname{div}\mathbf{E}\,dV=\frac1{\varepsilon_0}\int_V q\,dV.</math>

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

<math display="block">-\int_V \operatorname{div}(\operatorname{grad}\varphi)\,dV = \frac1{\varepsilon_0} \int_V q\,dV.</math>

Since this holds for all regions , we must have

<math display="block">\operatorname{div}(\operatorname{grad}\varphi) = -\frac 1 {\varepsilon_0}q</math>

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 in a region are functions that make the Dirichlet energy functional stationary:

<math display="block"> E(f) = \frac{1}{2} \int_U \lVert \nabla f \rVert^2 \,dx.</math>

To see this, suppose is a function, and is a function that vanishes on the boundary of . Then:

<math display="block">\left. \frac{d}{d\varepsilon}\right|_{\varepsilon = 0} E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, dx = -\int_U u \, \Delta f\, dx </math>

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

Coordinate expressions

Two dimensions

The Laplace operator in two dimensions is given by:

In Cartesian coordinates,

<math display="block">\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}</math>

where and are the standard Cartesian coordinates of the -plane.

In polar coordinates,

<math display="block">\begin{align}

\Delta f &= \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} \\

&= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2},

\end{align}</math>

where represents the radial distance and the angle.

Three dimensions

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

In Cartesian coordinates,

<math display="block">\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.</math>

In cylindrical coordinates,

<math display="block">\Delta f = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2 },</math>

where <math>\rho</math> represents the radial distance, the azimuth angle and the height.

In spherical coordinates:

<math display="block">\Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},</math>

<math display="block">\Delta f = \frac{1}{r} \frac{\partial^2}{\partial r^2} (r f) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},</math>

by expanding the first and second term, these expressions read

<math display="block">\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2 \sin \theta} \left(\cos \theta \frac{\partial f}{\partial \theta} + \sin \theta \frac{\partial^2 f}{\partial \theta^2} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},</math>

where represents the azimuthal angle and the zenith angle or co-latitude. In particular, the above is equivalent to

<math>\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\Delta_{S^2} f ,</math>

where <math>\Delta_{S^2}f</math> is the Laplace-Beltrami operator on the unit sphere.

In general curvilinear coordinates ():

<math display="block">\Delta = \nabla \xi^m \cdot \nabla \xi^n \frac{\partial^2}{\partial \xi^m \, \partial \xi^n} + \nabla^2 \xi^m \frac{\partial}{\partial \xi^m } = g^{mn} \left(\frac{\partial^2}{\partial\xi^m \, \partial\xi^n} - \Gamma^{l}_{mn}\frac{\partial}{\partial\xi^l} \right),</math>

where summation over the repeated indices is implied,

is the inverse metric tensor and are the Christoffel symbols for the selected coordinates.

N dimensions

In arbitrary curvilinear coordinates <math>(\xi^1,\dots,\xi^N)</math> on <math>\mathbf R^N</math>, the Laplacian can be written in terms of the inverse metric tensor <math>g^{ij}</math> as

\Delta f

\frac{1}{\sqrt{|g|

\frac{\partial}{\partial \xi^i}

\left(

\sqrt{|g|}\,g^{ij}\frac{\partial f}{\partial \xi^j}

\right),

\qquad |g|=\det(g_{ij}).

</math>

This is the Euclidean special case of the Laplace–Beltrami operator.

In spherical coordinates on <math>\mathbf R^N</math>, write

x=r\omega, \qquad r=|x|>0,\quad \omega\in S^{N-1}

</math>

where <math>S^{N-1}</math> is the unit (N–1)-sphere in <math>\mathbf R^N.</math> Then the Laplacian decomposes into radial and angular parts:

\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-1f,

</math>

or equivalently

\Delta f

\frac{1}{r^{N-1\frac{\partial}{\partial r}

\left(r^{N-1}\frac{\partial f}{\partial r}\right)

\frac{1}{r^2}\Delta_{S^{N-1f,

</math>

where <math>\Delta_{S^{N-1</math> is the Laplace–Beltrami operator on <math>S^{N-1}</math>, often called the spherical Laplacian.

This decomposition is the starting point for separation of variables in Laplace's equation. If one seeks solutions of the form

u(r,\omega)=R(r)Y(\omega),

</math>

then the angular factor must satisfy the eigenvalue equation

-\Delta_{S^{N-1Y=\lambda Y.

</math>

The eigenvalues are

\lambda_\ell=\ell(\ell+N-2), \qquad \ell=0,1,2,\dots,

</math>

and the corresponding eigenfunctions are the spherical harmonics of degree <math>\ell</math> on <math>S^{N-1}</math>.

Substituting <math>u(r,\omega)=R(r)Y(\omega)</math> into <math>\Delta u=0</math> gives the radial equation

r^2R(r)+(N-1)rR'(r)-\ell(\ell+N-2)R(r)=0.

</math>

For <math>N\ge 3</math>, its solutions are

R(r)=Ar^\ell+Br^{-\ell-(N-2)},

</math>

while in the exceptional case <math>N=2</math> and <math>\ell=0</math> one obtains

R(r)=A+B\log r.

</math>

These give the classical solid harmonics.

In particular, if <math>f(x)=F(r)</math> is radial, then the angular term vanishes and

\Delta f

F(r)+\frac{N-1}{r}F'(r)

\frac{1}{r^{N-1\frac{d}{dr}\left(r^{N-1}F'(r)\right).

</math>

Thus every radial harmonic function on an annulus in <math>\mathbf R^N</math> has the form

F(r)=

\begin{cases}

A+Br^{2-N}, & N\ne 2,\\[4pt]

A+B\log r, & N=2.

\end{cases}

</math>

As a consequence, the spherical Laplacian of a function on <math>S^{N-1}</math> may be computed by extending the function to <math>\mathbf R^N\setminus\{0\}</math> so that it is constant along rays (that is, homogeneous of degree <math>0</math>) and then applying the ordinary Laplacian.

Euclidean invariance

The Laplacian is equivariant under pullback by every Euclidean transformation. More precisely, if

is a Euclidean isometry of <math>\mathbf R^n</math>, with <math>U\in O(n)</math> and <math>a\in \mathbf R^n</math>, then for every <math>f\in C^2(\mathbf R^n)</math>,

<math display="block">\Delta(f\circ g)=(\Delta f)\circ g.</math>

Thus the Laplacian commutes with translations and with orthogonal transformations, hence in particular with rotations and reflections.

In two dimensions, this says that for every angle <math>\theta</math> and every translation vector <math>(a,b)</math>,

<math display="block">\Delta\bigl(f(x\cos\theta-y\sin\theta+a,\;x\sin\theta+y\cos\theta+b)\bigr)

(\Delta f)(x\cos\theta-y\sin\theta+a,\;x\sin\theta+y\cos\theta+b).</math>

Equivalently, <math>\Delta</math> is invariant under the natural action of the Euclidean group <math>E(n)=O(n)\ltimes \mathbf R^n</math> on functions on <math>\mathbf R^n</math>.

More generally, on homogeneous spaces such as spheres, the Laplace–Beltrami operator is obtained from the quadratic Casimir of the acting Lie group, and on a compact Lie group with a bi-invariant metric the Laplacian is the image of the Casimir element of the Lie algebra.

Properties

The Laplace operator has several basic structural properties that make it the prototype of an elliptic operator.

Linearity and ellipticity

The Laplace operator is linear:

<math display="block">\Delta(af+bg)=a\,\Delta f+b\,\Delta g</math>

for all functions <math>f</math> and <math>g</math> and scalars <math>a</math> and <math>b</math>.

For the sign convention used in this article,

<math display="block">\Delta=\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2},</math>

the principal symbol of <math>\Delta</math> is

<math display="block">\sigma_2(\Delta)(\xi)=-|\xi|^2,</math>

which is nonzero for every <math>\xi \ne 0</math>. Thus <math>\Delta</math> is an elliptic differential operator.

Green's identities and formal self-adjointness

If <math>\Omega \subset \mathbf R^n</math> is a bounded <math>C^1</math> domain and <math>u,v\in C^2(\bar\Omega)</math>, then Green's identities give

<math display="block">\int_\Omega u\,\Delta v\,dx

-\int_\Omega \nabla u\cdot \nabla v\,dx

\int_{\partial\Omega} u\,\frac{\partial v}{\partial \nu}\,dS,</math>

and

<math display="block">\int_\Omega u\,\Delta v\,dx

\int_\Omega v\,\Delta u\,dx

\int_{\partial\Omega}\left(u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}\right)\,dS.</math>

In particular, if the boundary term vanishes (for example, for compactly supported functions), then

<math display="block">\int_\Omega u\,\Delta v\,dx=\int_\Omega v\,\Delta u\,dx,</math>

so the Laplacian is formally self-adjoint. In particular, harmonic functions are smooth, and in fact real analytic.

This representation makes several basic features of the Laplacian transparent. The symbol depends only on <math>|\xi|</math>, reflecting the rotational invariance of the operator, and it is nonzero for <math>\xi\ne 0</math>, reflecting ellipticity.

Under other common Fourier-transform conventions, the factor <math>4\pi^2</math> is redistributed, but the essential statement remains the same: the Fourier transform diagonalizes the Laplacian.

<math display="block">(-\Delta)^{\alpha/2}I_\alpha f=f.</math>

A related family of operators is given by the Bessel potentials. For <math>s\in\mathbf{R}</math>, the Bessel potential operator is defined by

<math display="block">\mathcal{F}\big((I-\Delta)^{-s/2}f\big)(\xi)=(1+4\pi^2|\xi|^2)^{-s/2}\widehat{f}(\xi).</math>

The associated function spaces are the Bessel potential spaces <math>H^{s,p}(\mathbf{R}^n)</math>. Riesz and Bessel potentials are closely related smoothing operators, but Bessel potentials involve <math>I-\Delta</math> rather than <math>-\Delta</math> and therefore behave better at low frequencies. The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field, returning a vector quantity. When computed in orthonormal Cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

The vector Laplacian of a vector field <math> \mathbf{A} </math> is defined as

<math display="block"> \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}). </math>

This definition can be seen as the Helmholtz decomposition of the vector Laplacian.

In Cartesian coordinates, this reduces to the much simpler expression

<math display="block"> \nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z), </math>

where <math>A_x</math>, <math>A_y</math>, and <math>A_z</math> are the components of the vector field <math>\mathbf{A}</math>, and <math> \nabla^2 </math> just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product.

For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates.

Generalization

The Laplacian of any tensor field <math>\mathbf{T}</math> ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor:

<math display="block">\nabla ^2\mathbf{T} = (\nabla \cdot \nabla) \mathbf{T}.</math>

For the special case where <math>\mathbf{T}</math> is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form.

If <math>\mathbf{T}</math> is a vector (a tensor of first degree), the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the Jacobian matrix shown below for the gradient of a vector:

<math display="block">\nabla \mathbf{T}= (\nabla T_x, \nabla T_y, \nabla T_z) = \begin{bmatrix}

T_{xx} & T_{xy} & T_{xz} \\

T_{yx} & T_{yy} & T_{yz} \\

T_{zx} & T_{zy} & T_{zz}

\end{bmatrix} ,

\text{ where } T_{uv} \equiv \frac{\partial T_u}{\partial v}.</math>

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:

<math display="block"> \mathbf{A} \cdot \nabla \mathbf{B}

= \begin{bmatrix} A_x & A_y & A_z \end{bmatrix} \nabla \mathbf{B}

= \begin{bmatrix} \mathbf{A} \cdot \nabla B_x & \mathbf{A} \cdot \nabla B_y & \mathbf{A} \cdot \nabla B_z \end{bmatrix}.</math>

This identity is a coordinate dependent result, and is not general.

Use in physics

An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow:

<math display="block">\rho \left(\frac{\partial \mathbf{v{\partial t}+ ( \mathbf{v} \cdot \nabla ) \mathbf{v}\right)=\rho \mathbf{f}-\nabla p +\mu\left(\nabla ^2 \mathbf{v}\right),</math>

where the term with the vector Laplacian of the velocity field <math>\mu\left(\nabla ^2 \mathbf{v}\right)</math> represents the viscous stresses in the fluid.

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:

<math display="block">\nabla^2 \mathbf{E} - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E{\partial t^2} = 0.</math>

This equation can also be written as:

<math display="block">\Box\, \mathbf{E} = 0,</math>

where <math display="block">\Box\equiv\frac{1}{c^2} \frac{\partial^2}{\partial t^2}-\nabla^2,</math> is the d'Alembertian, used in the Klein–Gordon equation.

Semigroup and heat kernel

The Laplace operator generates the heat semigroup . If solves the heat equation

<math display="block">\partial_t u = \Delta u</math>

on <math>\mathbf{R}^n</math> with initial data , then

<math display="block">u(x,t)=(e^{t\Delta}f)(x)=\int_{\mathbf{R}^n}\Gamma_t(x-y)f(y)\,dy,</math>

where

<math display="block">\Gamma_t(x)=\frac{1}{(4\pi t)^{n/2e^{-|x|^2/4t}</math>

is the Euclidean heat kernel.

On bounded domains and on Riemannian manifolds, the same construction defines a heat semigroup whose integral kernel is again called the heat kernel. Its short-time asymptotic behaviour encodes geometric and spectral information about the underlying space.