Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols ⁠\(\nabla\cdot\nabla\)⁠, \(\nabla^2\) (where \(\nabla\) is the nabla operator), or ⁠\(\Delta\)⁠. 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 Δf (p) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f (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 Δf = 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 ⁠\(\mathbf R^n\)⁠.

The Laplace operator is a second-order differential operator in the n-dimensional Euclidean space, defined as the divergence (⁠\(\nabla \cdot\)⁠) of the gradient (⁠\(\nabla f\)⁠). Thus if \(f\) is a twice-differentiable real-valued function, then the Laplacian of \(f\) is the real-valued function defined by:

where the latter notations derive from formally writing: \[\nabla = \left ( \frac{\partial }{\partial x_1} , \ldots , \frac{\partial }{\partial x_n} \right ).\] Explicitly, the Laplacian of f is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates x_i:

As a second-order differential operator, the Laplace operator maps C functions to C functions for k ≥ 2. It is a linear operator Δ : C(R) → C(R), or more generally, an operator Δ : C(Ω) → C(Ω) for any open set Ω ⊆ R.

Alternatively, the Laplace operator can be defined as: \[\nabla^2 f(\vec{x}) = \lim_{R \rightarrow 0} \frac{2n}{R^2} (f_{\text{shell}_R} - f(\vec{x})) = \lim_{R \rightarrow 0} \frac{2n}{A_{n-1} R^{1+n}} \int_{\text{shell}_R} f(\vec{r}) - f(\vec{x}) d r^{n-1}\] where \(n\) is the dimension of the space, \(f_{\text{shell}_R}\) is the average value of \(f\) on the surface of an n-sphere of radius ⁠\(R\)⁠, \(\textstyle \int_{\text{shell}_R} f(\vec{r}) d r^{n-1}\) is the surface integral over an n-sphere of radius ⁠\(R\)⁠, and \(A_{n-1}\) is the hypervolume of the boundary of a unit n-sphere.

There is no single standard sign convention for the Laplace operator. In Euclidean coordinates, one common convention is \[\Delta=\nabla\cdot\nabla=\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2},\] so that for every smooth compactly supported function \(\varphi\), \[\int_{\mathbf R^n}\overline{\varphi(x)}\,\Delta\varphi(x)\,dx = -\int_{\mathbf R^n} |\nabla \varphi(x)|^2\,dx,\] and hence \(\Delta\) is negative semidefinite on \(L^2\).

Another common convention inserts a minus sign and defines instead \[\Delta=-\nabla\cdot\nabla=-\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2},\] so that the Laplacian is nonnegative.

Both conventions occur in the literature, and authors usually state explicitly which one they are using. In this article, unless otherwise noted, \(\Delta\) denotes the Euclidean Laplacian \[\Delta=\nabla\cdot\nabla=\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2}.\]

In the physical theory of diffusion, the Laplace operator arises naturally in the mathematical description of equilibrium. Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary ∂V (also called S) of any smooth region V is zero, provided there is no source or sink within V: \[\int_{S} \nabla u \cdot \mathbf{n}\, dS = 0,\] where n is the outward unit normal to the boundary of V. By the divergence theorem, \[\int_V \operatorname{div} \nabla u\, dV = \int_{S} \nabla u \cdot \mathbf{n}\, dS = 0.\]

Since this holds for all smooth regions V, one can show that it implies: \[\operatorname{div} \nabla u = \Delta u = 0.\] The left-hand side of this equation is the Laplace operator, and the entire equation Δu = 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.

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

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

If φ denotes the electrostatic potential associated to a charge distribution q, then the charge distribution itself is given by the negative of the Laplacian of φ: \[q = -\varepsilon_0 \Delta\varphi,\] where ε₀ is the electric constant.

This is a consequence of Gauss's law. Indeed, if V is any smooth region with boundary ∂V, then by Gauss's law the flux of the electrostatic field E across the boundary is proportional to the charge enclosed: \[\int_{\partial V} \mathbf{E}\cdot \mathbf{n}\, dS = \int_V \operatorname{div}\mathbf{E}\,dV=\frac1{\varepsilon_0}\int_V q\,dV.\] where the first equality is due to the divergence theorem. Since the electrostatic field is the (negative) gradient of the potential, this gives: \[-\int_V \operatorname{div}(\operatorname{grad}\varphi)\,dV = \frac1{\varepsilon_0} \int_V q\,dV.\]

Since this holds for all regions V, we must have \[\operatorname{div}(\operatorname{grad}\varphi) = -\frac 1 {\varepsilon_0}q\]

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.

Another motivation for the Laplacian appearing in physics is that solutions to Δf = 0 in a region U are functions that make the Dirichlet energy functional stationary: \[E(f) = \frac{1}{2} \int_U \lVert \nabla f \rVert^2 \,dx.\]

To see this, suppose f : U → R is a function, and u : U → R is a function that vanishes on the boundary of U. Then: \[\left. \frac{d}{d\varepsilon}\right|_{\varepsilon = 0} E(f+\varepsilon u) = \int_U \nabla f \cdot \nabla u \, dx = -\int_U u \, \Delta f\, dx\] where the last equality follows using Green's first identity. This calculation shows that if Δf = 0, then E is stationary around f. Conversely, if E is stationary around f, then Δf = 0 by the fundamental lemma of calculus of variations.

The Laplace operator in two dimensions is given by:

In Cartesian coordinates, \[\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\] where x and y are the standard Cartesian coordinates of the xy-plane.

In polar coordinates, \[\begin{align} \Delta f &= \frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial f}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2} \\ &= \frac{\partial^2 f}{\partial r^2} + \frac{1}{r} \frac{\partial f}{\partial r} + \frac{1}{r^2} \frac{\partial^2 f}{\partial \theta^2}, \end{align}\] where r represents the radial distance and θ the angle.

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

In Cartesian coordinates, \[\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.\]

In cylindrical coordinates, \[\Delta f = \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho} \right) + \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2 },\] where \(\rho\) represents the radial distance, φ the azimuth angle and z the height.

In spherical coordinates: \[\Delta f = \frac{1}{r^2} \frac{\partial}{\partial r} \left(r^2 \frac{\partial f}{\partial r} \right) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},\] or \[\Delta f = \frac{1}{r} \frac{\partial^2}{\partial r^2} (r f) + \frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial f}{\partial \theta} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},\] by expanding the first and second term, these expressions read \[\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r}+\frac{1}{r^2 \sin \theta} \left(\cos \theta \frac{\partial f}{\partial \theta} + \sin \theta \frac{\partial^2 f}{\partial \theta^2} \right) + \frac{1}{r^2 \sin^2 \theta} \frac{\partial^2 f}{\partial \varphi^2},\] where φ represents the azimuthal angle and θ the zenith angle or co-latitude. In particular, the above is equivalent to \(\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\Delta_{S^2} f ,\) where \(\Delta_{S^2}f\) is the Laplace-Beltrami operator on the unit sphere.

In general curvilinear coordinates (ξ, ξ, ξ): \[\Delta = \nabla \xi^m \cdot \nabla \xi^n \frac{\partial^2}{\partial \xi^m \, \partial \xi^n} + \nabla^2 \xi^m \frac{\partial}{\partial \xi^m } = g^{mn} \left(\frac{\partial^2}{\partial\xi^m \, \partial\xi^n} - \Gamma^{l}_{mn}\frac{\partial}{\partial\xi^l} \right),\] where summation over the repeated indices is implied, g is the inverse metric tensor and Γ _mn are the Christoffel symbols for the selected coordinates.

In arbitrary curvilinear coordinates \((\xi^1,\dots,\xi^N)\) on \(\mathbf R^N\), the Laplacian can be written in terms of the inverse metric tensor \(g^{ij}\) as \[\Delta f = \frac{1}{\sqrt{|g|}} \frac{\partial}{\partial \xi^i} \left( \sqrt{|g|}\,g^{ij}\frac{\partial f}{\partial \xi^j} \right), \qquad |g|=\det(g_{ij}).\] This is the Euclidean special case of the Laplace-Beltrami operator.

In spherical coordinates on \(\mathbf R^N\), write \[x=r\omega, \qquad r=|x|>0,\quad \omega\in S^{N-1}\] where \(S^{N-1}\) is the unit (N–1)-sphere in \(\mathbf R^N.\) Then the Laplacian decomposes into radial and angular parts: \[\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{N-1}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\Delta_{S^{N-1}}f,\] or equivalently \[\Delta f = \frac{1}{r^{N-1}}\frac{\partial}{\partial r} \left(r^{N-1}\frac{\partial f}{\partial r}\right) + \frac{1}{r^2}\Delta_{S^{N-1}}f,\] where \(\Delta_{S^{N-1}}\) is the Laplace-Beltrami operator on \(S^{N-1}\), often called the spherical Laplacian.

This decomposition is the starting point for separation of variables in Laplace's equation. If one seeks solutions of the form \[u(r,\omega)=R(r)Y(\omega),\] then the angular factor must satisfy the eigenvalue equation \[-\Delta_{S^{N-1}}Y=\lambda Y.\] The eigenvalues are \[\lambda_\ell=\ell(\ell+N-2), \qquad \ell=0,1,2,\dots,\] and the corresponding eigenfunctions are the spherical harmonics of degree \(\ell\) on \(S^{N-1}\).

Substituting \(u(r,\omega)=R(r)Y(\omega)\) into \(\Delta u=0\) gives the radial equation \[r^2R''(r)+(N-1)rR'(r)-\ell(\ell+N-2)R(r)=0.\] For \(N\ge 3\), its solutions are \[R(r)=Ar^\ell+Br^{-\ell-(N-2)},\] while in the exceptional case \(N=2\) and \(\ell=0\) one obtains \[R(r)=A+B\log r.\] These give the classical solid harmonics.

In particular, if \(f(x)=F(r)\) is radial, then the angular term vanishes and \[\Delta f = F''(r)+\frac{N-1}{r}F'(r) = \frac{1}{r^{N-1}}\frac{d}{dr}\left(r^{N-1}F'(r)\right).\] Thus every radial harmonic function on an annulus in \(\mathbf R^N\) has the form \[F(r)= \begin{cases} A+Br^{2-N}, & N\ne 2,\\[4pt] A+B\log r, & N=2. \end{cases}\]

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

The Laplacian is equivariant under pullback by every Euclidean transformation. More precisely, if \[g(x)=Ux+a\] is a Euclidean isometry of \(\mathbf R^n\), with \(U\in O(n)\) and \(a\in \mathbf R^n\), then for every \(f\in C^2(\mathbf R^n)\), \[\Delta(f\circ g)=(\Delta f)\circ g.\] Thus the Laplacian commutes with translations and with orthogonal transformations, hence in particular with rotations and reflections.

In two dimensions, this says that for every angle \(\theta\) and every translation vector \((a,b)\), \[\Delta\bigl(f(x\cos\theta-y\sin\theta+a,\;x\sin\theta+y\cos\theta+b)\bigr) = (\Delta f)(x\cos\theta-y\sin\theta+a,\;x\sin\theta+y\cos\theta+b).\]

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

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

From the viewpoint of Lie theory, the rotational invariance of the Laplacian is reflected in the action of the orthogonal group \(O(n)\). 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 \[\Delta_{\mathbf R^3} = \frac{1}{r}\frac{\partial^2}{\partial r^2}(r\cdot)-\frac{J^2}{r^2} = \frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}+\frac{1}{r^2}\Delta_{S^2},\] where \(J^2\) is the quadratic Casimir of \(\mathfrak{so}(3)\); equivalently, \(\Delta_{S^2}=-J^2\) up to convention.

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.

The Laplace operator is linear: \[\Delta(af+bg)=a\,\Delta f+b\,\Delta g\] for all functions \(f\) and \(g\) and scalars \(a\) and \(b\).

For the sign convention used in this article, \[\Delta=\sum_{j=1}^n \frac{\partial^2}{\partial x_j^2},\] the principal symbol of \(\Delta\) is \[\sigma_2(\Delta)(\xi)=-|\xi|^2,\] which is nonzero for every \(\xi \ne 0\). Thus \(\Delta\) is an elliptic differential operator.

If \(\Omega \subset \mathbf R^n\) is a bounded \(C^1\) domain and \(u,v\in C^2(\bar\Omega)\), then Green's identities give \[\int_\Omega u\,\Delta v\,dx = -\int_\Omega \nabla u\cdot \nabla v\,dx + \int_{\partial\Omega} u\,\frac{\partial v}{\partial \nu}\,dS,\] and \[\int_\Omega u\,\Delta v\,dx = \int_\Omega v\,\Delta u\,dx + \int_{\partial\Omega}\left(u\frac{\partial v}{\partial \nu}-v\frac{\partial u}{\partial \nu}\right)\,dS.\] In particular, if the boundary term vanishes (for example, for compactly supported functions), then \[\int_\Omega u\,\Delta v\,dx=\int_\Omega v\,\Delta u\,dx,\] so the Laplacian is formally self-adjoint. Taking \(u=v\) gives the energy identity \[\int_\Omega u\,\Delta u\,dx = -\int_\Omega |\nabla u|^2\,dx + \int_{\partial\Omega} u\,\frac{\partial u}{\partial \nu}\,dS,\] which underlies uniqueness results for boundary value problems.

A twice continuously differentiable function \(u\) is called harmonic if \(\Delta u=0\), subharmonic if \(\Delta u\ge 0\), and superharmonic if \(\Delta u\le 0\).

If \(u\) is harmonic in an open set \(\Omega\) and \(B_r(x)\subset \Omega\), then \(u(x)\) equals both the average of \(u\) over the ball \(B_r(x)\) and the average of \(u\) over the sphere \(\partial B_r(x)\). This is the mean value property for harmonic functions.

A nonconstant harmonic function cannot attain an interior maximum or minimum. Consequently, if \(\Omega\) is bounded and \(u\in C^2(\Omega)\cap C(\bar\Omega)\) is harmonic, then \[\max_{\bar{\Omega}} u = \max_{\partial\Omega} u, \qquad \min_{\bar{\Omega}} u = \min_{\partial\Omega} u.\] These are the maximum principle and minimum principle for harmonic functions.