# Laplace operator

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{{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](/source/mathematics), the '''Laplace operator''' or '''Laplacian''' is a [differential operator](/source/differential_operator) given by the [divergence](/source/divergence) of the [gradient](/source/gradient) of a [scalar function](/source/Scalar_field) on [Euclidean space](/source/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](/source/Del)), or {{tmath| \Delta }}. In a [Cartesian coordinate system](/source/Cartesian_coordinate_system), the Laplacian is given by the sum of second [partial derivative](/source/partial_derivative)s of the function with respect to each [independent variable](/source/independent_variable). In other [coordinate systems](/source/coordinate_systems), such as [cylindrical](/source/cylindrical_coordinates) and [spherical coordinates](/source/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](/source/Pierre-Simon_de_Laplace) (1749–1827), who first applied the operator to the study of [celestial mechanics](/source/celestial_mechanics): the Laplacian of the [gravitational potential](/source/gravitational_potential) due to a given mass density distribution is a constant multiple of that density distribution. Solutions of [Laplace's equation](/source/Laplace's_equation) {{math|1=Δ''f'' = 0}} are called [harmonic function](/source/harmonic_function)s and represent the possible [gravitational potential](/source/gravitational_potential)s in regions of [vacuum](/source/vacuum).

The Laplacian occurs in many [differential equations](/source/differential_equations) describing physical phenomena. [Poisson's equation](/source/Poisson's_equation) describes  [electric](/source/electric_potential) and [gravitational potential](/source/gravitational_potential)s; the [diffusion equation](/source/diffusion_equation) describes [heat](/source/heat_equation) and [fluid flow](/source/fluid_mechanics); the [wave equation](/source/wave_equation) describes [wave propagation](/source/wave_equation); and the [Schrödinger equation](/source/Schr%C3%B6dinger_equation) describes the [wave function](/source/wave_function) in [quantum mechanics](/source/quantum_mechanics). In [image processing](/source/image_processing) and [computer vision](/source/computer_vision), the Laplacian operator has been used for various tasks, such as [blob](/source/blob_detection) and [edge detection](/source/edge_detection). The Laplacian is the simplest [elliptic operator](/source/elliptic_operator) and is at the core of [Hodge theory](/source/Hodge_theory) as well as the results of [de Rham cohomology](/source/de_Rham_cohomology). It is also essentially the [infinitesimal generator](/source/infinitesimal_generator_(stochastic_processes)) of standard [Brownian motion](/source/Brownian_motion) on {{tmath| \mathbf R^n }}.

== Definition ==
The Laplace operator is a [second-order differential operator](/source/Second-order_differential_equation) in the ''n''-dimensional [Euclidean space](/source/Euclidean_space), defined as the [divergence](/source/divergence) ({{tmath| \nabla \cdot }}) of the [gradient](/source/gradient) ({{tmath| \nabla f }}). Thus if <math>f</math> is a [twice-differentiable](/source/derivative) [real-valued function](/source/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 derivative](/source/partial_derivative)s in the [Cartesian coordinates](/source/Cartesian_coordinates) {{math|''x<sub>i</sub>''}}:
{{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}}''](/source/Continuously_differentiable)}} 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'''<sup>''n''</sup>) → ''C''{{i sup|''k''−2}}('''R'''<sup>''n''</sup>)}}, or more generally, an operator {{math|Δ : ''C''{{i sup|''k''}}(Ω) → ''C''{{i sup|''k''−2}}(Ω)}} for any [open set](/source/open_set) {{math|Ω ⊆ '''R'''<sup>''n''</sup>}}.

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](/source/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](/source/Unit_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](/source/physics) theory of [diffusion](/source/diffusion), the Laplace operator arises naturally in the mathematical description of [equilibrium](/source/Diffusion_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](/source/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](/source/unit_normal) to the boundary of {{math|''V''}}. By the [divergence theorem](/source/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](/source/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](/source/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](/source/electrostatic_potential) associated to a [charge distribution](/source/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''<sub>0</sub>}} is the [electric constant](/source/electric_constant).

This is a consequence of [Gauss's law](/source/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](/source/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](/source/gravitational_potential) is the [mass distribution](/source/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](/source/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](/source/Dirichlet_energy) [functional](/source/functional_(mathematics)) [stationary](/source/stationary_point):
<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](/source/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](/source/fundamental_lemma_of_calculus_of_variations).

== Coordinate expressions ==

=== Two dimensions ===
The Laplace operator in two dimensions is given by:

In '''[Cartesian coordinates](/source/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](/source/Cartesian_coordinates) of the {{math|''xy''}}-plane.

In '''[polar coordinates](/source/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](/source/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](/source/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](/source/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](/source/azimuthal_angle) and {{math|''θ''}} the [zenith angle](/source/zenith_angle) or [co-latitude](/source/colatitude). 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](/source/Laplace%E2%80%93Beltrami_operator) on the unit sphere. 
<!---**************************************************************-->

In general '''[curvilinear coordinates](/source/curvilinear_coordinates)''' ({{math|''ξ''<sup>1</sup>, ''ξ''<sup>2</sup>, ''ξ''<sup>3</sup>}}):
<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](/source/Einstein_summation_convention),
{{math|''g<sup>mn</sup>''}} is the inverse [metric tensor](/source/metric_tensor) and  {{math|Γ''<sup>l</sup> <sub>mn</sub>''}} are the [Christoffel symbols](/source/Christoffel_symbols) for the selected coordinates.

=== ''N'' dimensions ===
In arbitrary [curvilinear coordinates](/source/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](/source/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](/source/Laplace%E2%80%93Beltrami_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](/source/n-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](/source/Laplace%E2%80%93Beltrami_operator) on <math>S^{N-1}</math>, often called the [spherical Laplacian](/source/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](/source/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 harmonic](/source/solid_harmonic)s.<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](/source/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](/source/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](/source/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](/source/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](/source/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](/source/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](/source/harmonic_function)'' if <math>\Delta u=0</math>, ''[subharmonic](/source/subharmonic_function)'' 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](/source/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](/source/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](/source/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](/source/maximum_principle) and [Harnack's inequality](/source/Harnack's_inequality), is discussed in more detail in [Laplace's equation](/source/Laplace's_equation) and [harmonic function](/source/harmonic_function).

== Fourier transform ==
For sufficiently regular functions on <math>\mathbf{R}^n</math>, the Laplacian is particularly simple after applying the [Fourier transform](/source/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](/source/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](/source/Dirichlet_boundary_condition) or [Neumann](/source/Neumann_boundary_condition) 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](/source/Helmholtz_equation).<ref name="Hunter" />

More generally, on a compact [Riemannian manifold](/source/Riemannian_manifold), the [Laplace–Beltrami operator](/source/Laplace%E2%80%93Beltrami_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](/source/spherical_harmonics).<ref name="Canzani" />

=== Fractional Laplacian ===
A nonlocal generalization of the Laplace operator is given by the ''[fractional Laplacian](/source/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](/source/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](/source/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](/source/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 potential](/source/Bessel_potential)s''. 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 space](/source/Bessel_potential_space)s <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](/source/differential_operator) defined over a [vector field](/source/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](/source/scalar_field) and returns a scalar quantity, the vector Laplacian applies to a [vector field](/source/vector_field), returning a vector quantity. When computed in [orthonormal](/source/orthonormal) [Cartesian coordinates](/source/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](/source/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](/source/Helmholtz_decomposition) of the vector Laplacian.

In [Cartesian coordinate](/source/Cartesian_coordinate)s, 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](/source/Vector_triple_product).

For expressions of the vector Laplacian in other coordinate systems see [Del in cylindrical and spherical coordinates](/source/Del_in_cylindrical_and_spherical_coordinates).

=== Generalization ===
The Laplacian of any [tensor field](/source/tensor_field) <math>\mathbf{T}</math> ("tensor" includes scalar and vector) is defined as the [divergence](/source/divergence) of the [gradient](/source/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](/source/scalar_(mathematics)) (a tensor of degree zero), the [Laplacian](/source/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](/source/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](/source/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](/source/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](/source/Navier-Stokes_equations) for a [Newtonian](/source/Newtonian_fluid) [incompressible flow](/source/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](/source/velocity) field <math>\mu\left(\nabla ^2 \mathbf{v}\right)</math> represents the [viscous](/source/viscosity) [stress](/source/Stress_(physics))es in the fluid.

Another example is the wave equation for the electric field that can be derived from [Maxwell's equations](/source/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](/source/d'Alembert_operator), used in the [Klein–Gordon equation](/source/Klein%E2%80%93Gordon_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](/source/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](/source/Sobolev_space) 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 equation](/source/parabolic_partial_differential_equation)s and underlies Gaussian smoothing and [scale-space](/source/scale-space_representation) methods in [image processing](/source/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 manifold](/source/Riemannian_manifold)s, 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](/source/Dirichlet_energy) makes sense, which is the theory of [Dirichlet form](/source/Dirichlet_form)s.  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](/source/Laplace%E2%80%93Beltrami_operator)''' defined on a [Riemannian manifold](/source/Riemannian_manifold). The Laplace–Beltrami operator, when applied to a function, is the [trace](/source/trace_(linear_algebra)) ({{math|tr}}) of the function's [Hessian](/source/Hessian_matrix):
<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](/source/metric_tensor). The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on [tensor field](/source/tensor_field)s, by a similar formula.

Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the [exterior derivative](/source/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](/source/codifferential), which can also be expressed in terms of the [Hodge star](/source/Hodge_star_operator) 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 form](/source/differential_form)s {{mvar|α}} by
<math display="block">\Delta \alpha = \delta d \alpha + d \delta \alpha .</math>

This is known as the '''[Laplace–de Rham operator](/source/Laplace%E2%80%93Beltrami_operator)''', which is related to the Laplace–Beltrami operator by the [Weitzenböck identity](/source/Weitzenb%C3%B6ck_identity).

=== D'Alembertian ===
The Laplacian can be generalized in certain ways to [non-Euclidean](/source/non-Euclidean) spaces, where it may be [elliptic](/source/elliptic_operator), [hyperbolic](/source/hyperbolic_operator), or [ultrahyperbolic](/source/ultrahyperbolic_operator).

In [Minkowski space](/source/Minkowski_space) the [Laplace–Beltrami operator](/source/Laplace%E2%80%93Beltrami_operator) becomes the [D'Alembert operator](/source/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](/source/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](/source/particle_physics). The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the [wave equation](/source/wave_equation)s, and it is also part of the [Klein–Gordon equation](/source/Klein%E2%80%93Gordon_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](/source/Natural_units)}} in order to simplify the equation.

The d'Alembert operator generalizes to a hyperbolic operator on [pseudo-Riemannian manifold](/source/pseudo-Riemannian_manifold)s.

== See also ==
* [Laplace–Beltrami operator](/source/Laplace%E2%80%93Beltrami_operator), generalization to submanifolds in Euclidean space and Riemannian and pseudo-Riemannian manifold.
* The [Laplacian in differential geometry](/source/Laplace_operators_in_differential_geometry).
* The [discrete Laplace operator](/source/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](/source/image_processing) and [computer vision](/source/computer_vision) (see the [Laplacian of Gaussian](/source/Laplacian_of_Gaussian), [blob detector](/source/blob_detection), and [scale space](/source/scale_space)).
* The [list of formulas in Riemannian geometry](/source/list_of_formulas_in_Riemannian_geometry) contains expressions for the Laplacian in terms of Christoffel symbols.
* [Weyl's lemma (Laplace equation)](/source/Weyl's_lemma_(Laplace_equation)).
* [Earnshaw's theorem](/source/Earnshaw's_theorem) which shows that stable static gravitational, electrostatic or magnetic suspension is impossible.
* [Del in cylindrical and spherical coordinates](/source/Del_in_cylindrical_and_spherical_coordinates).
* Other situations in which a Laplacian is defined are: [analysis on fractals](/source/analysis_on_fractals), [time scale calculus](/source/time_scale_calculus) and [discrete exterior calculus](/source/discrete_exterior_calculus).
* [Nodal line conjecture](/source/Nodal_line_conjecture), regarding the location of the nodal line of the second Dirichlet eigenfunction
* [Schrödinger operator](/source/Schr%C3%B6dinger_operator)
* [Paneitz operator](/source/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

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