Laplace operator

This article is about themathematical operator.For the integral transform, seeLaplace transform.For the Laplace probability distribution, seeLaplace distribution.For graph theoretical notion, seeLaplacian matrix.

Inmathematics,theLaplace operatororLaplacianis adifferential operatorgiven by thedivergenceof thegradientof ascalar functiononEuclidean space.It is usually denoted by the symbols$\nabla \cdot \nabla$,$\nabla ^{2}$(where$\nabla$is thenabla operator), or$\Delta$.In aCartesian coordinate system,the Laplacian is given by the sum of secondpartial derivativesof the function with respect to eachindependent variable.In othercoordinate systems,such ascylindricalandspherical coordinates,the Laplacian also has a useful form. Informally, the LaplacianΔf (p)of a functionfat a pointpmeasures by how much the average value offover small spheres or balls centered atpdeviates fromf (p).

The Laplace operator is named after the French mathematicianPierre-Simon de Laplace(1749–1827), who first applied the operator to the study ofcelestial mechanics:the Laplacian of thegravitational potentialdue to a given mass density distribution is a constant multiple of that density distribution. Solutions ofLaplace's equationΔf= 0are calledharmonic functionsand represent the possiblegravitational potentialsin regions ofvacuum.

The Laplacian occurs in manydifferential equationsdescribing physical phenomena.Poisson's equationdescribeselectricandgravitational potentials;thediffusion equationdescribesheatandfluid flow;thewave equationdescribeswave propagation;and theSchrödinger equationdescribes thewave functioninquantum mechanics.Inimage processingandcomputer vision,the Laplacian operator has been used for various tasks, such asblobandedge detection.The Laplacian is the simplestelliptic operatorand is at the core ofHodge theoryas well as the results ofde Rham cohomology.

Definition

The Laplace operator is asecond-order differential operatorin then-dimensionalEuclidean space,defined as thedivergence($\nabla \cdot$) of thegradient($\nabla f$). Thus if$f$is atwice-differentiablereal-valued function,then the Laplacian of$f$is the real-valued function defined by:

$$\Delta f=\nabla ^{2}f=\nabla \cdot \nabla f$$1

where the latter notations derive from formally writing: $$\nabla =\left({\frac {\partial }{\partial x_{1}}},\ldots,{\frac {\partial }{\partial x_{n}}}\right).$$ Explicitly, the Laplacian offis thus the sum of all theunmixedsecondpartial derivativesin theCartesian coordinatesx_i:

$$\Delta f=\sum _{i=1}^{n}{\frac {\partial ^{2}f}{\partial x_{i}^{2}}}$$2

As a second-order differential operator, the Laplace operator mapsC^kfunctions toC^k−2functions fork≥ 2.It is a linear operatorΔ:C^k(Rⁿ) →C^k−2(Rⁿ),or more generally, an operatorΔ:C^k(Ω) →C^k−2(Ω)for anyopen setΩ ⊆Rⁿ.

Alternatively, the Laplace operator can be defined as:$\nabla ^{2}f({\overrightarrow {x}})=\lim _{R\rightarrow 0}{\frac {2n}{R^{2}}}(f_{shell_{R}}-f({\overrightarrow {x}}))=\lim _{R\rightarrow 0}{\frac {2n}{A_{n-1}R^{1+n}}}\int _{shell_{R}}f({\overrightarrow {r}})-f({\overrightarrow {x}})dr^{n-1}$

Where$n$is the dimension of the space,$f_{shell_{R}}$is the average value of$f$on the surface of an-sphereof radius R,$\int _{shell_{R}}f({\overrightarrow {r}})dr^{n-1}$is the surface integral over an-sphereof radius R, and$A_{n-1}$is thehypervolume of the boundary of a unit n-sphere.

Motivation

Diffusion

In thephysicaltheory ofdiffusion,the Laplace operator arises naturally in the mathematical description ofequilibrium.Specifically, ifuis the density at equilibrium of some quantity such as a chemical concentration, then thenet fluxofuthrough the boundary∂V(also calledS) of any smooth regionVis zero, provided there is no source or sink withinV: $$\int _{S}\nabla u\cdot \mathbf {n} \,dS=0,$$ wherenis the outwardunit normalto the boundary ofV.By thedivergence theorem, $$\int _{V}\operatorname {div} \nabla u\,dV=\int _{S}\nabla u\cdot \mathbf {n} \,dS=0.$$

Since this holds for all smooth regionsV,one can show that it implies: $$\operatorname {div} \nabla u=\Delta u=0.$$ The left-hand side of this equation is the Laplace operator, and the entire equationΔu= 0is known asLaplace'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 thediffusion equation.This interpretation of the Laplacian is also explained by the following fact about averages.

Averages

Given a twice continuously differentiable function$f:\mathbb {R} ^{n}\to \mathbb {R}$and a point$p\in \mathbb {R} ^{n}$,the average value of$f$over the ball with radius$h$centered at$p$is: $${\overline {f}}_{B}(p,h)=f(p)+{\frac {\Delta f(p)}{2(n+2)}}h^{2}+o(h^{2})\quad {\text{for}}\;\;h\to 0$$

Similarly, the average value of$f$over the sphere (the boundary of a ball) with radius$h$centered at$p$is: $${\overline {f}}_{S}(p,h)=f(p)+{\frac {\Delta f(p)}{2n}}h^{2}+o(h^{2})\quad {\text{for}}\;\;h\to 0.$$

Density associated with a potential

Ifφdenotes theelectrostatic potentialassociated to acharge distributionq,then the charge distribution itself is given by the negative of the Laplacian ofφ: $$q=-\varepsilon _{0}\Delta \varphi,$$ whereε₀is theelectric constant.

This is a consequence ofGauss's law.Indeed, ifVis any smooth region with boundary∂V,then by Gauss's law the flux of the electrostatic fieldEacross the boundary is proportional to the charge enclosed: $$\int _{\partial V}\mathbf {E} \cdot \mathbf {n} \,dS=\int _{V}\operatorname {div} \mathbf {E} \,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.$$ where the first equality is due to thedivergence theorem.Since the electrostatic field is the (negative) gradient of the potential, this gives: $$-\int _{V}\operatorname {div} (\operatorname {grad} \varphi )\,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.$$

Since this holds for all regionsV,we must have $$\operatorname {div} (\operatorname {grad} \varphi )=-{\frac {1}{\varepsilon _{0}}}q$$

The same approach implies that the negative of the Laplacian of thegravitational potentialis themass 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 solvingPoisson's equation.

Energy minimization

Another motivation for the Laplacian appearing in physics is that solutions toΔf= 0in a regionUare functions that make theDirichlet energyfunctionalstationary: $$E(f)={\frac {1}{2}}\int _{U}\lVert \nabla f\rVert ^{2}\,dx.$$

To see this, supposef:U→Ris a function, andu:U→Ris a function that vanishes on the boundary ofU.Then: $$\left.{\frac {d}{d\varepsilon }}\right|_{\varepsilon =0}E(f+\varepsilon u)=\int _{U}\nabla f\cdot \nabla u\,dx=-\int _{U}u\,\Delta f\,dx$$

where the last equality follows usingGreen's first identity.This calculation shows that ifΔf= 0,thenEis stationary aroundf.Conversely, ifEis stationary aroundf,thenΔf= 0by thefundamental lemma of calculus of variations.

Coordinate expressions

Two dimensions

The Laplace operator in two dimensions is given by:

InCartesian coordinates, $$\Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}$$ wherexandyare the standardCartesian coordinatesof thexy-plane.

Inpolar coordinates, $${\begin{aligned}\Delta f&={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}\\&={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}},\end{aligned}}$$ whererrepresents the radial distance andθthe angle.

Three dimensions

See also:Del in cylindrical and spherical coordinates

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

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

Incylindrical coordinates, $$\Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}},$$ where$\rho$represents the radial distance,φthe azimuth angle andzthe height.

Inspherical coordinates: $$\Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},$$ or $$\Delta f={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(rf)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},$$ by expanding the first and second term, these expressions read $$\Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}\sin \theta }}\left(\cos \theta {\frac {\partial f}{\partial \theta }}+\sin \theta {\frac {\partial ^{2}f}{\partial \theta ^{2}}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},$$ whereφrepresents theazimuthal angleandθthezenith angleorco-latitude.In particular, the above is equivalent to

$\Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{2}}f,$

where$\Delta _{S^{2}}f$is theLaplace-Beltrami operatoron the unit sphere.

In generalcurvilinear coordinates(ξ¹,ξ²,ξ³): $$\Delta =\nabla \xi ^{m}\cdot \nabla \xi ^{n}{\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}+\nabla ^{2}\xi ^{m}{\frac {\partial }{\partial \xi ^{m}}}=g^{mn}\left({\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}-\Gamma _{mn}^{l}{\frac {\partial }{\partial \xi ^{l}}}\right),$$

wheresummation over the repeated indices is implied, g^mnis the inversemetric tensorandΓ^l_mnare theChristoffel symbolsfor the selected coordinates.

Ndimensions

In arbitrarycurvilinear coordinatesinNdimensions (ξ¹,...,ξ^N), we can write the Laplacian in terms of the inversemetric tensor,$g^{ij}$: $$\Delta ={\frac {1}{\sqrt {\det g}}}{\frac {\partial }{\partial \xi ^{i}}}\left({\sqrt {\det g}}g^{ij}{\frac {\partial }{\partial \xi ^{j}}}\right),$$ from theVoss-Weylformulafor thedivergence.

Inspherical coordinates inNdimensions,with the parametrizationx=rθ∈R^Nwithrrepresenting a positive real radius andθan element of theunit sphereS^N−1, $$\Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {N-1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{N-1}}f$$ whereΔ_S^N−1is theLaplace–Beltrami operatoron the(N− 1)-sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as: $${\frac {1}{r^{N-1}}}{\frac {\partial }{\partial r}}\left(r^{N-1}{\frac {\partial f}{\partial r}}\right).$$

As a consequence, the spherical Laplacian of a function defined onS^N−1⊂R^Ncan be computed as the ordinary Laplacian of the function extended toR^N∖{0}so that it is constant along rays, i.e.,homogeneousof degree zero.

Euclidean invariance

The Laplacian is invariant under allEuclidean transformations:rotationsandtranslations.In two dimensions, for example, this means that: $$\Delta (f(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b))=(\Delta f)(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b)$$ for allθ,a,andb.In arbitrary dimensions, $$\Delta (f\circ \rho )=(\Delta f)\circ \rho$$ wheneverρis a rotation, and likewise: $$\Delta (f\circ \tau )=(\Delta f)\circ \tau$$ wheneverτis a translation. (More generally, this remains true whenρis anorthogonal transformationsuch as areflection.)

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 theory

See also:Hearing the shape of a drumandDirichlet eigenvalue

Thespectrumof the Laplace operator consists of alleigenvaluesλfor which there is a correspondingeigenfunctionfwith: $$-\Delta f=\lambda f.$$

This is known as theHelmholtz equation.

IfΩis a bounded domain inRⁿ,then the eigenfunctions of the Laplacian are anorthonormal basisfor theHilbert spaceL²(Ω).This result essentially follows from thespectral theoremoncompactself-adjoint operators,applied to the inverse of the Laplacian (which is compact, by thePoincaré inequalityand theRellich–Kondrachov theorem).It can also be shown that the eigenfunctions areinfinitely differentiablefunctions.More generally, these results hold for theLaplace–Beltrami operatoron any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of anyelliptic operatorwith smooth coefficients on a bounded domain. WhenΩis then-sphere,the eigenfunctions of the Laplacian are thespherical harmonics.

Vector Laplacian

Thevector Laplace operator,also denoted by$\nabla ^{2}$,is adifferential operatordefined over avector field.The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to ascalar fieldand returns a scalar quantity, the vector Laplacian applies to avector field,returning a vector quantity. When computed inorthonormalCartesian coordinates,the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.

Thevector Laplacianof avector field$\mathbf {A}$is defined as $$\nabla ^{2}\mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla \times (\nabla \times \mathbf {A} ).$$ This definition can be seen as theHelmholtz decompositionof the vector Laplacian.

InCartesian coordinates,this reduces to the much simpler form as $$\nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z}),$$ where$A_{x}$,$A_{y}$,and$A_{z}$are the components of the vector field$\mathbf {A}$,and$\nabla ^{2}$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; seeVector triple product.

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

Generalization

The Laplacian of anytensor field$\mathbf {T}$( "tensor" includes scalar and vector) is defined as thedivergenceof thegradientof the tensor: $$\nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T}.$$

For the special case where$\mathbf {T}$is ascalar(a tensor of degree zero), theLaplaciantakes on the familiar form.

If$\mathbf {T}$is a vector (a tensor of first degree), the gradient is acovariant derivativewhich 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 theJacobian matrixshown below for the gradient of a vector: $$\nabla \mathbf {T} =(\nabla T_{x},\nabla T_{y},\nabla T_{z})={\begin{bmatrix}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{bmatrix}},{\text{ where }}T_{uv}\equiv {\frac {\partial T_{u}}{\partial v}}.$$

And, in the same manner, adot 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: $$\mathbf {A} \cdot \nabla \mathbf {B} ={\begin{bmatrix}A_{x}&A_{y}&A_{z}\end{bmatrix}}\nabla \mathbf {B} ={\begin{bmatrix}\mathbf {A} \cdot \nabla B_{x}&\mathbf {A} \cdot \nabla B_{y}&\mathbf {A} \cdot \nabla B_{z}\end{bmatrix}}.$$ This identity is a coordinate dependent result, and is not general.

Use in physics

An example of the usage of the vector Laplacian is theNavier-Stokes equationsfor aNewtonianincompressible flow: $$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\nabla ^{2}\mathbf {v} \right),$$ where the term with the vector Laplacian of thevelocityfield$\mu \left(\nabla ^{2}\mathbf {v} \right)$represents theviscousstressesin the fluid.

Another example is the wave equation for the electric field that can be derived fromMaxwell's equationsin the absence of charges and currents: $$\nabla ^{2}\mathbf {E} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0.$$

This equation can also be written as: $$\Box \,\mathbf {E} =0,$$ where$$\Box \equiv {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2},$$is theD'Alembertian,used in theKlein–Gordon equation.

Some properties

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

Let$u\colon \Omega \to \mathbb {R}$be a smooth function, and let$K\subset \Omega$be a connected compact set. If$u$is superharmonic, then, for every$x\in K$,we have $$u(x)\geq \inf _{\Omega }u+c\lVert u\rVert _{L^{1}(K)}\;,$$ for some constant$c>0$depending on$\Omega$and$K$.

Generalizations

A version of the Laplacian can be defined wherever theDirichlet energy functionalmakes sense, which is the theory ofDirichlet forms.For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.

Laplace–Beltrami operator

Main article:Laplace–Beltrami operator

The Laplacian also can be generalized to an elliptic operator called theLaplace–Beltrami operatordefined on aRiemannian manifold.The Laplace–Beltrami operator, when applied to a function, is thetrace(tr) of the function'sHessian: $$\Delta f=\operatorname {tr} {\big (}H(f){\big )}$$ where the trace is taken with respect to the inverse of themetric tensor.The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates ontensor fields,by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses theexterior derivative,in terms of which the "geometer's Laplacian" is expressed as $$\Delta f=\delta df.$$

Hereδis thecodifferential,which can also be expressed in terms of theHodge starand the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined ondifferential formsαby $$\Delta \ Alpha =\delta d\ Alpha +d\delta \ Alpha.$$

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

D'Alembertian

The Laplacian can be generalized in certain ways tonon-Euclideanspaces, where it may beelliptic,hyperbolic,orultrahyperbolic.

InMinkowski spacetheLaplace–Beltrami operatorbecomes theD'Alembert operator$\Box$or D'Alembertian: $$\square ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}.$$

It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under theisometry groupof 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-energyparticle physics.The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in thewave equations,and it is also part of theKlein–Gordon equation,which reduces to the wave equation in the massless case.

The additional factor ofcin the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, thexdirection were measured in meters while theydirection were measured in centimeters. Indeed, theoretical physicists usually work in units such thatc= 1in order to simplify the equation.

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

See also

• Laplace–Beltrami operator,generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold.
• TheLaplacian in differential geometry.
• Thediscrete Laplace operatoris a finite-difference analog of the continuous Laplacian, defined on graphs and grids.
• The Laplacian is a common operator inimage processingandcomputer vision(see theLaplacian of Gaussian,blob detector,andscale space).
• Thelist of formulas in Riemannian geometrycontains expressions for the Laplacian in terms of Christoffel symbols.
• Weyl's lemma (Laplace equation).
• Earnshaw's theoremwhich shows that stable static gravitational, electrostatic or magnetic suspension is impossible.
• Del in cylindrical and spherical coordinates.
• Other situations in which a Laplacian is defined are:analysis on fractals,time scale calculusanddiscrete exterior calculus.

References

• Evans, L. (1998),Partial Differential Equations,American Mathematical Society,ISBN978-0-8218-0772-9
• The Feynman Lectures on Physics Vol. II Ch. 12: Electrostatic Analogs
• Gilbarg, D.;Trudinger, N.(2001),Elliptic Partial Differential Equations of Second Order,Springer,ISBN978-3-540-41160-4.
• Schey, H. M. (1996),Div, Grad, Curl, and All That,W. W. Norton,ISBN978-0-393-96997-9.

Further reading

• The Laplacian - Richard Fitzpatrick 2006

External links

• "Laplace operator",Encyclopedia of Mathematics,EMS Press,2001 [1994]
• Weisstein, Eric W."Laplacian".MathWorld.
• Laplacian in polar coordinates derivation
• Laplace equations on the fractal cubes and Casimir effect