Laplace operator

{{short description|Differential operator in mathematics}} {{about|the mathematical operator|the integral transform|Laplace transform|other uses|List of things named after Pierre-Simon Laplace}} {{Calculus |Vector}}

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 {{tmath| \nabla\cdot\nabla }}, <math>\nabla^2</math> (where <math>\nabla</math> is the nabla operator), or {{tmath| \Delta }}. 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 {{math|Δ''f''&hairsp;(''p'')}} of a function {{math|''f''}} at a point {{math|''p''}} measures by how much the average value of {{math|''f''}} over small spheres or balls centered at {{math|''p''}} deviates from {{math|''f''&hairsp;(''p'')}}.

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 {{math|1=Δ''f'' = 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 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 {{tmath| \mathbf R^n }}.

== Definition == The Laplace operator is a second-order differential operator in the ''n''-dimensional Euclidean space, defined as the divergence ({{tmath| \nabla \cdot }}) of the gradient ({{tmath| \nabla f }}). 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: {{NumBlk||<math display="block">\Delta f = \nabla^2 f = \nabla \cdot \nabla f ,</math>|{{EqRef|1}}}} 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 {{math|''f''}} is thus the sum of all the ''unmixed'' second partial derivatives in the Cartesian coordinates {{math|''x_i''}}: {{NumBlk||<math display="block">\Delta f = \sum_{i=1}^n \frac {\partial^2 f}{\partial x^2_i}</math>|{{EqRef|2}}}}

As a second-order differential operator, the Laplace operator maps {{math|''C{{i sup|k}}''}} functions to {{math|''C''{{i sup|''k''−2}}}} functions for {{math|''k'' ≥ 2}}. It is a linear operator {{math|Δ : ''C''{{i sup|''k''}}('''R'''^''n'') → ''C''{{i sup|''k''−2}}('''R'''^''n'')}}, or more generally, an operator {{math|Δ : ''C''{{i sup|''k''}}(Ω) → ''C''{{i sup|''k''−2}}(Ω)}} for any open set {{math|Ω ⊆ '''R'''^''n''}}.

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 {{tmath| R }}, <math>\textstyle \int_{\text{shell}_R} f(\vec{r}) d r^{n-1}</math> is the surface integral over an {{math|''n''}}-sphere of radius {{tmath| R }}, and <math>A_{n-1}</math> is the hypervolume of the boundary of a unit {{math|''n''}}-sphere.<ref>{{cite journal |last=Styer |first=Daniel F. |date=2015-12-01 |title=The geometrical significance of the Laplacian |url=https://pubs.aip.org/aapt/ajp/article-abstract/83/12/992/1057202/The-geometrical-significance-of-the-Laplacian?redirectedFrom=fulltext |journal=American Journal of Physics |volume=83 |issue=12 |pages=992–997 |doi=10.1119/1.4935133 |bibcode=2015AmJPh..83..992S |issn=0002-9505 |archive-url=https://web.archive.org/web/20241010001312/https://pubs.aip.org/aapt/ajp/article-abstract/83/12/992/1057202/The-geometrical-significance-of-the-Laplacian?redirectedFrom=fulltext |archive-date=2024-10-10 |url-access=subscription |access-date=2024-10-10 |url-status=bot: unknown }}</ref>

== 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>.<ref name="Evans">{{cite book |last=Evans |first=L. C. |title=Partial Differential Equations |edition=2nd |publisher=American Mathematical Society |year=2010 |isbn=978-0-8218-4974-3}}</ref>

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.<ref>{{cite book |last=Rosenberg |first=Steven |title=The Laplacian on a Riemannian Manifold |series=London Mathematical Society Student Texts |volume=31 |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-46831-2}}</ref><ref>{{cite web |last=Donaldson |first=S. K. |author-link=Simon Donaldson |title=Lecture Notes for TCC Course ''Geometric Analysis'' |url=https://www.ma.imperial.ac.uk/~skdona/GEOMETRICANALYSIS.PDF |access-date=2026-03-21}}</ref>

Both conventions occur in the literature, and authors usually state explicitly which one they are using.<ref name="Hunter">{{cite web |last=Hunter |first=John K. |title=Chapter 2: Laplace's equation |url=https://www.math.ucdavis.edu/~hunter/pdes/ch2.pdf |access-date=2026-03-21}}</ref> 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.<ref>{{harvnb|Evans|1998|loc=§2.2}}</ref> Specifically, if {{math|''u''}} is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of {{math|''u''}} through the boundary {{math|∂''V''}} (also called {{math|''S''}}) of any smooth region {{math|''V''}} is zero, provided there is no source or sink within {{math|''V''}}: <math display="block">\int_{S} \nabla u \cdot \mathbf{n}\, dS = 0,</math> where {{math|'''n'''}} is the outward unit normal to the boundary of {{math|''V''}}. 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 {{math|''V''}}, 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 {{math|1=Δ''u'' = 0}} 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 {{tmath| p\in\R^n }}, the average value of <math>f </math> over the ball with radius <math>h</math> centered at <math>p</math> is:<ref>{{cite journal | last=Ovall | first=Jeffrey S. | date=2016-03-01 | title=The Laplacian and Mean and Extreme Values | url=http://web.pdx.edu/~jovall/PDF/LaplaceMeanValue.pdf | journal=The American Mathematical Monthly | volume=123 | issue=3 | pages=287–291 | doi=10.4169/amer.math.monthly.123.3.287 | s2cid=124943537 | archive-date=2024-10-07 | access-date=2020-07-26 | archive-url=https://web.archive.org/web/20241007160429/https://web.pdx.edu/~jovall/PDF/LaplaceMeanValue.pdf | url-status=dead }}</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 potential === If {{math|''φ''}} denotes the electrostatic potential associated to a charge distribution {{math|''q''}}, then the charge distribution itself is given by the negative of the Laplacian of {{math|''φ''}}: <math display="block">q = -\varepsilon_0 \Delta\varphi,</math> where {{math|''ε''₀}} is the electric constant.

This is a consequence of Gauss's law. Indeed, if {{math|''V''}} is any smooth region with boundary {{math|∂''V''}}, then by Gauss's law the flux of the electrostatic field {{math|'''E'''}} 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 {{mvar|V}}, 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 {{math|1=Δ''f'' = 0}} in a region {{math|''U''}} 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 {{math|''f'' : ''U'' → '''R'''}} is a function, and {{math|''u'' : ''U'' → '''R'''}} is a function that vanishes on the boundary of {{mvar|U}}. 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 {{math|1=Δ''f'' = 0}}, then {{math|''E''}} is stationary around {{math|''f''}}. Conversely, if {{math|''E''}} is stationary around {{math|''f''}}, then {{math|1=Δ''f'' = 0}} 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 {{mvar|x}} and {{mvar|y}} are the standard Cartesian coordinates of the {{math|''xy''}}-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 {{mvar|r}} represents the radial distance and {{mvar|θ}} the 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.

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, {{math|''φ''}} the azimuth angle and {{math|''z''}} 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> <!---**********PLEASE SEE THE DISCUSSION PAGE BEFORE CHANGING THIS.**********--> where {{math|''φ''}} represents the azimuthal angle and {{math|''θ''}} 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|''ξ''¹, ''ξ''², ''ξ''³}}): <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, {{math|''g^mn''}} is the inverse metric tensor and {{math|Γ''^l _mn''}} 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 <math display="block"> \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.<ref>{{cite book |last1=Courant |first1=Richard |last2=Hilbert |first2=David |title=Methods of Mathematical Physics, Volume I |publisher=Wiley-Interscience |year=1962}}</ref><ref>{{cite book |last1=Axler |first1=Sheldon |last2=Bourdon |first2=Paul |last3=Ramey |first3=Wade |title=Harmonic Function Theory |edition=2nd |publisher=Springer |year=2001 |isbn=978-0-387-95218-5}}</ref>

In '''spherical coordinates''' on <math>\mathbf R^N</math>, write <math display="block"> 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: <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> or equivalently <math display="block"> \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-1}}f, </math> where <math>\Delta_{S^{N-1}}</math> is the Laplace–Beltrami operator on <math>S^{N-1}</math>, often called the spherical Laplacian.<ref>{{cite book |last1=Axler |first1=Sheldon |last2=Bourdon |first2=Paul |last3=Ramey |first3=Wade |title=Harmonic Function Theory |edition=2nd |publisher=Springer |year=2001 |isbn=978-0-387-95218-5}}</ref>

This decomposition is the starting point for separation of variables in Laplace's equation. If one seeks solutions of the form <math display="block"> u(r,\omega)=R(r)Y(\omega), </math> then the angular factor must satisfy the eigenvalue equation <math display="block"> -\Delta_{S^{N-1}}Y=\lambda Y. </math> The eigenvalues are <math display="block"> \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>.<ref>{{cite book |last1=Courant |first1=Richard |last2=Hilbert |first2=David |title=Methods of Mathematical Physics, Volume I |publisher=Wiley-Interscience |year=1962}}</ref>

Substituting <math>u(r,\omega)=R(r)Y(\omega)</math> into <math>\Delta u=0</math> gives the radial equation <math display="block"> r^2R''(r)+(N-1)rR'(r)-\ell(\ell+N-2)R(r)=0. </math> For <math>N\ge 3</math>, its solutions are <math display="block"> 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 <math display="block"> R(r)=A+B\log r. </math> These give the classical solid harmonics.<ref>{{cite book |last1=Axler |first1=Sheldon |last2=Bourdon |first2=Paul |last3=Ramey |first3=Wade |title=Harmonic Function Theory |edition=2nd |publisher=Springer |year=2001 |isbn=978-0-387-95218-5}}</ref>

In particular, if <math>f(x)=F(r)</math> is radial, then the angular term vanishes and <math display="block"> \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 <math display="block"> 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.<ref>{{cite book |last1=Axler |first1=Sheldon |last2=Bourdon |first2=Paul |last3=Ramey |first3=Wade |title=Harmonic Function Theory |edition=2nd |publisher=Springer |year=2001 |isbn=978-0-387-95218-5}}</ref>

== Euclidean invariance == The Laplacian is equivariant under pullback by every Euclidean transformation. More precisely, if <math display="block">g(x)=Ux+a</math> 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.<ref name="Canzani">{{cite web |last=Canzani |first=Yaiza |title=Analysis on Manifolds via the Laplacian |url=https://www.math.mcgill.ca/toth/spectral%20geometry.pdf |access-date=2026-03-21}}</ref><ref>{{cite web |last=Freire |first=Alexandre |title=Invariance of the Laplace Operator |url=https://web.math.utk.edu/~freire/teaching/m435s11/invLaplacian.pdf |access-date=2026-03-21}}</ref>

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>.<ref name="Canzani"/>

More generally, among scalar linear differential operators on <math>\mathbf R^n</math> with constant coefficients, those that commute with all Euclidean isometries are exactly the polynomial expressions in the Laplacian: <math display="block">P(\Delta)=a_0+a_1\Delta+\cdots+a_m\Delta^m.</math> In this sense, the Laplacian generates the algebra of Euclidean-invariant scalar constant-coefficient differential operators.<ref name="Canzani"/>

From the viewpoint of Lie theory, the rotational invariance of the Laplacian is reflected in the action of the orthogonal group <math>O(n)</math>. In particular, the angular part of the Euclidean Laplacian is, up to sign convention, the quadratic Casimir operator of the rotation group. In three dimensions this appears in the spherical-coordinate decomposition <math display="block">\Delta_{\mathbf R^3} = \frac{1}{r}\frac{\partial^2}{\partial r^2}(r\cdot)-\frac{J^2}{r^2} = \frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\Delta_{S^2},</math> where <math>J^2</math> is the quadratic Casimir of <math>\mathfrak{so}(3)</math>; equivalently, <math>\Delta_{S^2}=-J^2</math> up to convention.<ref>{{cite web |last=Zuber |first=Jean-Bernard |title=Rotations of <math>\mathbf R^3</math>, the groups <math>SO(3)</math> and <math>SU(2)</math> |url=https://www.lpthe.jussieu.fr/~zuber/Cours/chap0_12e.pdf |access-date=2026-03-21}}</ref>

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.<ref>{{cite web |last1=Schlichtkrull |first1=Henrik |last2=Trapa |first2=Peter |last3=Vogan |first3=David A. Jr. |title=Laplacians on spheres |url=https://math.mit.edu/~dav/spheres.pdf |access-date=2026-03-21}}</ref><ref>{{cite web |last=Ren |first=Qiuyu |title=Spectral Properties of the Laplacian on Compact Lie Groups |url=https://math.berkeley.edu/~qiuyu/papers/Spectral_Properties_of_the_Laplacian_on_Compact_Lie_Groups.pdf |access-date=2026-03-21}}</ref>

== 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.<ref>{{cite web |last=Dyatlov |first=Semyon |title=Lecture notes for 18.155: distributions, elliptic regularity, Fourier integral operators, and wave propagation |url=https://math.mit.edu/~dyatlov/18.155/155-notes.pdf |access-date=2026-03-21}}</ref>

=== 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.<ref name="Hunter"/> Taking <math>u=v</math> gives the energy identity <math display="block">\int_\Omega u\,\Delta u\,dx = -\int_\Omega |\nabla u|^2\,dx + \int_{\partial\Omega} u\,\frac{\partial u}{\partial \nu}\,dS,</math> which underlies uniqueness results for boundary value problems.<ref name="Hunter"/>

=== Harmonic, subharmonic, and superharmonic functions === A twice continuously differentiable function <math>u</math> is called ''harmonic'' if <math>\Delta u=0</math>, ''subharmonic'' if <math>\Delta u\ge 0</math>, and ''superharmonic'' if <math>\Delta u\le 0</math>.<ref name="Hunter"/>

If <math>u</math> is harmonic in an open set <math>\Omega</math> and <math>B_r(x)\subset \Omega</math>, then <math>u(x)</math> equals both the average of <math>u</math> over the ball <math>B_r(x)</math> and the average of <math>u</math> over the sphere <math>\partial B_r(x)</math>. This is the mean value property for harmonic functions.<ref name="Hunter"/>

A nonconstant harmonic function cannot attain an interior maximum or minimum. Consequently, if <math>\Omega</math> is bounded and <math>u\in C^2(\Omega)\cap C(\bar\Omega)</math> is harmonic, then <math display="block">\max_{\bar{\Omega}} u = \max_{\partial\Omega} u, \qquad \min_{\bar{\Omega}} u = \min_{\partial\Omega} u.</math> These are the maximum principle and minimum principle for harmonic functions.<ref name="Hunter"/>

=== Regularity === Because the Laplacian is elliptic, solutions of equations involving <math>\Delta</math> are more regular than might initially be assumed. In particular, if <math>\Delta u</math> is locally square-integrable, then <math>u</math> is locally in the Sobolev space <math>H^2</math>.<ref>{{cite web |last=Dyatlov |first=Semyon |title=Lecture notes for 18.155: distributions, elliptic regularity, Fourier integral operators, and wave propagation |url=https://math.mit.edu/~dyatlov/18.155/155-notes.pdf |access-date=2026-03-21}}</ref> In particular, harmonic functions are smooth, and in fact real analytic.<ref name="Hunter"/> More generally, Weyl's lemma states that if <math>u</math> is a distributional solution of <math>\Delta u=0</math>, then <math>u</math> is smooth.<ref name="Hunter"/>

The corresponding qualitative theory of harmonic functions, including the maximum principle and Harnack's inequality, is discussed in more detail in Laplace's equation and harmonic function.

== Fourier transform == For sufficiently regular functions on <math>\mathbf{R}^n</math>, the Laplacian is particularly simple after applying the Fourier transform. With the convention <math display="block">\widehat{f}(\xi)=\int_{\mathbf{R}^n} f(x)e^{-2\pi i x\cdot \xi}\,dx,</math> one has <math display="block">\widehat{\partial_j f}(\xi)=2\pi i \xi_j \widehat{f}(\xi),</math> and therefore <math display="block">\widehat{\Delta f}(\xi)=-4\pi^2 |\xi|^2 \widehat{f}(\xi).</math> Thus the Laplacian is a Fourier multiplier with symbol <math>-4\pi^2|\xi|^2</math>.<ref name="TaoFourier">{{cite web |last=Tao |first=Terence |title=Fourier Transform |url=https://www.math.ucla.edu/~tao/preprints/fourier.pdf |access-date=2026-03-21}}</ref>

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.<ref name="TaoFourier" /> It also allows one to define functions of the Laplacian by functional calculus: for example, the heat semigroup corresponds to multiplication by <math display="block">e^{-4\pi^2 t|\xi|^2},</math> and, more generally, fractional powers of the Laplacian correspond to multiplication by <math display="block">(4\pi^2|\xi|^2)^{\alpha/2}.</math><ref name="TaoFourier" /><ref name="KwasnickiFrac">{{cite journal |last=Kwaśnicki |first=Mateusz |title=Ten equivalent definitions of the fractional Laplace operator |journal=Fractional Calculus and Applied Analysis |volume=20 |issue=1 |year=2017 |pages=7–51 |doi=10.1515/fca-2017-0002|arxiv=1507.07356 }}</ref>

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.<ref name="TaoFourier" />

== Spectral theory == {{see also|Hearing the shape of a drum|Dirichlet eigenvalue|Neumann eigenvalue}}

The spectral theory of the Laplacian depends strongly on the underlying space and on the boundary conditions imposed.

On <math>L^2(\mathbf{R}^n)</math>, the Fourier transform diagonalizes <math>-\Delta</math>, turning it into multiplication by <math>4\pi^2|\xi|^2</math>.<ref name="TaoFourier" /> It follows that the spectrum of <math>-\Delta</math> on <math>\mathbf{R}^n</math> is the interval <math display="block">[0,\infty),</math> and that this spectrum is purely continuous rather than discrete.<ref name="Hunter"/> In this setting, plane waves are generalized eigenfunctions of the Laplacian.<ref name="TaoFourier" />

If <math>\Omega\subset \mathbf{R}^n</math> is a bounded domain and one imposes boundary conditions such as Dirichlet or Neumann conditions, then the corresponding realization of the Laplacian is a self-adjoint operator with compact resolvent. Consequently its spectrum is discrete: there is a sequence of eigenvalues <math display="block">0\le \lambda_1\le \lambda_2\le \cdots \to \infty</math> (counted with multiplicity), and the associated eigenfunctions form an orthonormal basis of <math>L^2(\Omega)</math>.<ref name="Hunter" /> The eigenvalue equation <math display="block">-\Delta u=\lambda u</math> is the Helmholtz equation.<ref name="Hunter" />

More generally, on a compact Riemannian manifold, the Laplace–Beltrami operator likewise has discrete nonnegative spectrum, and its eigenfunctions form an orthonormal basis of <math>L^2(M)</math>.<ref name="Canzani"/> On the round sphere, these eigenfunctions are the spherical harmonics.<ref name="Canzani" />

=== Fractional Laplacian === A nonlocal generalization of the Laplace operator is given by the ''fractional Laplacian'' <math>(-\Delta)^{\alpha/2}</math>, where <math>0<\alpha<2</math>. For Schwartz functions on <math>\mathbf{R}^n</math>, it may be defined by its Fourier transform: <math display="block">\mathcal{F}\big((-\Delta)^{\alpha/2}f\big)(\xi)=(4\pi^2|\xi|^2)^{\alpha/2}\widehat{f}(\xi),</math> using the Fourier-transform convention above.<ref name="KwasnickiFrac" />

Equivalently, the fractional Laplacian can be defined by a singular integral: <math display="block">(-\Delta)^{\alpha/2}f(x)=c_{n,\alpha}\,\operatorname{PV}\!\int_{\mathbf{R}^n}\frac{f(x)-f(y)}{|x-y|^{n+\alpha}}\,dy,</math> where <math>\operatorname{PV}</math> denotes the Cauchy principal value.<ref name="KwasnickiFrac" /> Unlike the ordinary Laplacian, this is a nonlocal operator: its value at a point depends on the values of the function on all of <math>\mathbf{R}^n</math>.<ref name="KwasnickiFrac" />

The inverse of the fractional Laplacian is closely related to the ''Riesz potential''. For <math>0<\alpha<n</math>, the Riesz potential of order <math>\alpha</math> is convolution with the kernel <math>c_{n,\alpha}|x|^{\alpha-n}</math>: <math display="block">I_\alpha f(x)=c_{n,\alpha}\int_{\mathbf{R}^n}\frac{f(y)}{|x-y|^{n-\alpha}}\,dy.</math> Where both sides are defined, one has<ref name="KwasnickiFrac" /><ref>{{cite web |last1=Bucur |first1=Claudia |last2=Valdinoci |first2=Enrico |title=What is the fractional Laplacian? A comparative review with new results |url=https://www.stt.msu.edu/~mcubed/fracLaplacian.pdf |access-date=2026-03-21}}</ref> <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>.<ref name="Demkowicz">{{cite web |last=Demkowicz |first=Leszek F. |title=Lecture Notes on Energy Spaces |url=https://users.oden.utexas.edu/~leszek/classes/CSE393/book.pdf |access-date=2026-03-21}}</ref> 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.<ref name="Demkowicz" />

== Vector Laplacian == The '''vector Laplace operator''', also denoted by {{tmath| \nabla^2 }}, is a differential operator defined over a vector field.<ref>{{cite web |url=http://mathworld.wolfram.com/VectorLaplacian.html |title=Vector Laplacian |author=MathWorld }}</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.

=== 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 == {{main|Heat kernel}} The Laplace operator generates the ''heat semigroup'' {{math|(''e''{{sup|''t''Δ}}){{sub|''t'' ≥ 0}}}}. If {{math|''u''(''x'', ''t'')}} solves the heat equation <math display="block">\partial_t u = \Delta u</math> on <math>\mathbf{R}^n</math> with initial data {{math|''u''(''x'',0){{=}}''f''(''x'')}}, 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/2}}e^{-|x|^2/4t}</math> is the Euclidean heat kernel.<ref name="Davies">{{cite book |last=Davies |first=E. B. |title=Heat Kernels and Spectral Theory |publisher=Cambridge University Press |year=1989}}</ref><ref name="Evans"/> The kernels satisfy <math display="block">\Gamma_{s+t}=\Gamma_s * \Gamma_t,</math> equivalently <math display="block">e^{(s+t)\Delta}=e^{s\Delta}e^{t\Delta}.</math> On <math>L^2(\mathbf{R}^n)</math> this is a strongly continuous contraction semigroup whose generator is the Laplacian; more generally, the heat semigroup acts contractively on {{math|''L''{{sup|''p''}}}} for {{math|1=1 ≤ ''p'' ≤ ∞}}.<ref name="Davies" /><ref name="Evans" />

A basic feature of the heat semigroup is its ''smoothing'' effect: for every {{math|''t'' > 0}}, the function {{math|''e''{{sup|''t''Δ}}''f''}} is smoother than the initial data {{math|''f''}}. In Euclidean space this is reflected in derivative estimates such as <math display="block"> \|\nabla^N e^{t\Delta}f\|_{L^p}\le C_N t^{-N/2}\|f\|_{L^p}, </math> and, more generally, in Sobolev estimates of the form <math display="block"> \|e^{t\Delta}f\|_{H^{s,p}}\le C\, t^{-s/2}\|f\|_{L^p}. </math> This instantaneous regularization is important in the theory of parabolic partial differential equations and underlies Gaussian smoothing and scale-space methods in image processing.<ref name="Evans" /><ref>{{cite journal |last=Lindeberg |first=Tony |title=Scale-space theory: A basic tool for analysing structures at different scales |journal=Journal of Applied Statistics |volume=21 |issue=1–2 |year=1994 |pages=225–270}}</ref>

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.<ref name="Davies" />

== Generalizations == 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 operator === {{main article|Laplace–Beltrami operator}}

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 ({{math|tr}}) 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 {{mvar|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 {{mvar|α}} 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'Alembertian === 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 {{math|''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 {{mvar|x}} direction were measured in meters while the {{mvar|y}} direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that {{math|1=''c'' = 1}} in order to simplify the equation.

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

== See also == * Laplace–Beltrami operator, generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold. * 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. * Del in cylindrical and spherical coordinates. * Other situations in which a Laplacian is defined are: analysis on fractals, time scale calculus and discrete exterior calculus. * Nodal line conjecture, regarding the location of the nodal line of the second Dirichlet eigenfunction * Schrödinger operator * Paneitz operator

== Notes == {{reflist|20em}}

== References == * {{citation |last=Evans |first=L. |title=Partial Differential Equations |publisher=American Mathematical Society |year=1998 |isbn=978-0-8218-0772-9 }} * [https://feynmanlectures.caltech.edu/II_12.html The Feynman Lectures on Physics Vol. II Ch. 12: Electrostatic Analogs] * {{citation |author2-link=Neil Trudinger |first1=D. |last1=Gilbarg |first2=N. |last2=Trudinger |title=Elliptic Partial Differential Equations of Second Order |year=2001 |publisher=Springer |isbn=978-3-540-41160-4 }}. * {{citation |last=Schey |first=H. M. |title=Div, Grad, Curl, and All That |publisher=W. W. Norton |year=1996 |isbn=978-0-393-96997-9 }}.

== Further reading == * [http://farside.ph.utexas.edu/teaching/em/lectures/node23.html The Laplacian - Richard Fitzpatrick 2006]

== External links == * {{springer|title=Laplace operator|id=p/l057510}} * {{MathWorld | urlname=Laplacian | title=Laplacian}} * [http://ramanujan.math.trinity.edu/rdaileda/teach/s12/m3357/lectures/lecture_3_27_2_short.pdf Laplacian in polar coordinates derivation] * [https://link.springer.com/article/10.1140/epjs/s11734-021-00317-4 Laplace equations on the fractal cubes and Casimir effect]

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Category:Differential operators Category:Elliptic partial differential equations Category:Fourier analysis Operator Category:Harmonic functions Category:Linear operators in calculus Category:Multivariable calculus