Laplace operator

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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\cdot\nabla</math>, <math>\nabla^2</math> (where <math>\nabla</math> is the nabla operator), or <math>\Delta</math>. 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 Template:Math of a function Template:Math at a point Template:Math measures by how much the average value of Template:Math over small spheres or balls centered at Template:Math deviates from Template:Math.

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 Template:Math 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.

DefinitionEdit

The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence (<math>\nabla \cdot</math>) of the gradient (<math>\nabla f</math>). 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: Template:NumBlk 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 Template:Math is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates Template:Math: Template:NumBlk

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

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_{shell_R} - f(\vec{x})) = \lim_{R \rightarrow 0} \frac{2n}{A_{n-1} R^{1+n}} \int_{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_{shell_R} </math> is the average value of <math>f</math> on the surface of an n-sphere of radius <math>R</math>, <math>\int_{shell_R} f(\vec{r}) d r^{n-1}</math> is the surface integral over an n-sphere of radius <math>R</math>, and <math>A_{n-1}</math> is the hypervolume of the boundary of a unit n-sphere.<ref>Template:Cite journal</ref>

Analytic and geometric LaplaciansEdit

There are two conflicting conventions as to how the Laplace operator is defined:

The "analytic" Laplacian, which could be characterized in <math>\R^n</math> as

<math display="block">\Delta=\nabla^2=\sum_{j=1}^n\Big(\frac{\partial}{\partial x_j}\Big)^2,</math> which is negative-definite in the sense that <math display="block">\int_{\R^n}\overline{\varphi(x)}\Delta\varphi(x)\,dx=-\int_{\R^n}|\nabla\varphi(x)|^2\,dx<0</math> for any smooth compactly supported function <math>\varphi\in C^\infty_c(\R^n)</math> which is not identically zero);

The "geometric", positive-definite Laplacian defined by

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

MotivationEdit

DiffusionEdit

In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium.<ref>Template:Harvnb</ref> Specifically, if Template:Math is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of Template:Math through the boundary Template:Math (also called Template:Math) of any smooth region Template:Math is zero, provided there is no source or sink within Template:Math: <math display="block">\int_{S} \nabla u \cdot \mathbf{n}\, dS = 0,</math> where Template:Math is the outward unit normal to the boundary of Template:Math. 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 Template:Math, 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 Template:Math 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.

AveragesEdit

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

If Template:Math denotes the electrostatic potential associated to a charge distribution Template:Math, then the charge distribution itself is given by the negative of the Laplacian of Template:Math: <math display="block">q = -\varepsilon_0 \Delta\varphi,</math> where Template:Math is the electric constant.

This is a consequence of Gauss's law. Indeed, if Template:Math is any smooth region with boundary Template:Math, then by Gauss's law the flux of the electrostatic field Template:Math 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 Template:Mvar, 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 minimizationEdit

Another motivation for the Laplacian appearing in physics is that solutions to Template:Math in a region Template:Math 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 Template:Math is a function, and Template:Math is a function that vanishes on the boundary of Template:Mvar. 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 Template:Math, then Template:Math is stationary around Template:Math. Conversely, if Template:Math is stationary around Template:Math, then Template:Math by the fundamental lemma of calculus of variations.

Coordinate expressionsEdit

Two dimensionsEdit

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 Template:Mvar and Template:Mvar are the standard Cartesian coordinates of the Template:Math-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 Template:Mvar represents the radial distance and Template:Mvar the angle.

Three dimensionsEdit

Template:See also 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, Template:Math the azimuth angle and Template:Math 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> or <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 Template:Math represents the azimuthal angle and Template:Math 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 (Template:Math): <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, Template:Math is the inverse metric tensor and Template:Math are the Christoffel symbols for the selected coordinates.

Template:Mvar dimensionsEdit

In arbitrary curvilinear coordinates in Template:Math dimensions (Template:Math), we can write the Laplacian in terms of the inverse metric tensor, <math> g^{ij} </math>: <math display="block">\Delta = \frac 1{\sqrt{\det g}}\frac{\partial}{\partial\xi^i} \left( \sqrt{\det g} \,g^{ij} \frac{\partial}{\partial \xi^j}\right) ,</math> from the Voss-Weyl formula<ref>Archived at Ghostarchive Template:Cbignore and the Wayback Machine Template:Cbignore: {{#invoke:citation/CS1|citation |CitationClass=web }}Template:Cbignore</ref> for the divergence.

In spherical coordinates in Template:Mvar dimensions, with the parametrization Template:Math with Template:Mvar representing a positive real radius and Template:Mvar an element of the unit sphere Template:Math, <math display="block"> \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</math> where Template:Math is the Laplace–Beltrami operator on the Template:Math-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as: <math display="block">\frac{1}{r^{N-1}} \frac{\partial}{\partial r} \left(r^{N-1} \frac{\partial f}{\partial r} \right).</math>

As a consequence, the spherical Laplacian of a function defined on Template:Math can be computed as the ordinary Laplacian of the function extended to Template:Math so that it is constant along rays, i.e., homogeneous of degree zero.

Euclidean invarianceEdit

The Laplacian is invariant under all Euclidean transformations: rotations and translations. In two dimensions, for example, this means that: <math display="block">\Delta ( f(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)) = (\Delta f)(x\cos\theta - y\sin\theta + a, x\sin\theta + y\cos\theta + b)</math> for all θ, a, and b. In arbitrary dimensions, <math display="block">\Delta (f\circ\rho) =(\Delta f)\circ \rho</math> whenever ρ is a rotation, and likewise: <math display="block">\Delta (f\circ\tau) =(\Delta f)\circ \tau</math> whenever τ is a translation. (More generally, this remains true when ρ is an orthogonal transformation such as a reflection.)

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.

Spectral theoryEdit

Template:See also The spectrum of the Laplace operator consists of all eigenvalues Template:Math for which there is a corresponding eigenfunction Template:Math with: <math display="block">-\Delta f = \lambda f.</math>

This is known as the Helmholtz equation.

If Template:Math is a bounded domain in Template:Math, then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space Template:Math. 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).<ref>Template:Harvnb</ref> It can also be shown that the eigenfunctions are infinitely differentiable functions.<ref>Template:Harvnb</ref> 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 Template:Math is the [[N-sphere|Template:Mvar-sphere]], the eigenfunctions of the Laplacian are the spherical harmonics.

Vector LaplacianEdit

The vector Laplace operator, also denoted by <math>\nabla^2</math>, is a differential operator defined over a vector field.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.

GeneralizationEdit

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 physicsEdit

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.

Some propertiesEdit

First of all, we say that a smooth function <math>u \colon \Omega \subset \mathbb R^N \to \mathbb R</math> is superharmonic whenever <math>-\Delta u \geq 0</math>.

Let <math>u \colon \Omega \to \mathbb R</math> be a smooth function, and let <math>K \subset \Omega</math> be a connected compact set. If <math>u</math> is superharmonic, then, for every <math>x \in K</math>, we have <math display="block"> u(x) \geq \inf_\Omega u + c\lVert u \rVert_{L^1(K)} \;, </math> for some constant <math>c > 0</math> depending on <math>\Omega</math> and <math>K</math>. <ref>Template:Cite book</ref>

GeneralizationsEdit

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.

Laplace–Beltrami operatorEdit

Template:Main article

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 (Template:Math) of the function's Hessian: <math display="block">\Delta f = \operatorname{tr}\big(H(f)\big)</math> 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 <math display="block"> \Delta f = \delta d f .</math>

Here Template:Mvar 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 Template:Mvar by <math display="block">\Delta \alpha = \delta d \alpha + d \delta \alpha .</math>

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

D'AlembertianEdit

The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic, hyperbolic, or ultrahyperbolic.

In Minkowski space the Laplace–Beltrami operator becomes the D'Alembert operator <math>\Box</math> or D'Alembertian: <math display="block">\square = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2} - \frac{\partial^2}{\partial z^2}.</math>

It is the generalization 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 Template:Math 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 Template:Mvar direction were measured in meters while the Template:Mvar direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that Template:Math in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds.

NotesEdit

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ReferencesEdit

External linksEdit

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