Laplace Operator

The Laplace operator, or Laplacian, is a second-order differential operator that maps a scalar function to the sum of its second spatial derivatives in orthogonal Cartesian coordinates. Usually denoted by $\Delta$ or $\nabla^2$, it is a central operator in partial differential equations, describing equilibrium potentials, diffusion, and other spatial processes. Its geometric generalization acts on functions on curved spaces, while discrete analogues act on grids and graphs. (web.stanford.edu)

Definition and basic properties

For a twice continuously differentiable function $f$ on an open subset of Euclidean space $\mathbb{R}^n$, the Laplacian is

Thus, in one dimension it is the ordinary second derivative; in three dimensions,

An equivalent definition is

the divergence of the gradient. It is also the trace of the Hessian matrix. These expressions describe the same operator in Cartesian coordinates. (web.stanford.edu)

The operator is linear:

for constant scalars $a,b$. It is invariant under translations and orthogonal changes of Cartesian coordinates: rotating the coordinate axes does not change the scalar quantity it computes. Unlike the Hessian, the Laplacian retains only the sum of directional second derivatives, not their full directional distribution. (math.mit.edu)

Some mathematical texts define the Laplacian with the opposite sign, as $-\operatorname{div}\nabla$. This entry uses $\Delta=\operatorname{div}\nabla$, so that $-\Delta$, with appropriate boundary conditions, is nonnegative. Sign conventions must be checked when comparing formulas, particularly in spectral theory and geometry. (arxiv.org)

Local interpretation

The Laplacian measures how a function differs from its nearby average. If $f$ is sufficiently smooth and $S^{n-1}$ is the unit sphere, Taylor expansion gives

Consequently, a positive Laplacian indicates that the average over a sufficiently small surrounding sphere exceeds the value at its center to leading order; a negative Laplacian indicates the reverse. (web.stanford.edu)

At an interior local minimum of a twice differentiable real-valued function, $\Delta f\geq0$; at an interior local maximum, $\Delta f\leq0$. The converses do not hold: the sum of second derivatives does not determine whether the point is an extremum. For example,

has $\Delta f=0$, although the origin is a saddle point. (web.stanford.edu)

A function satisfying

is called a harmonic function. Harmonic functions have an exact mean-value property on spheres and balls contained in their domain. On a connected domain, a nonconstant harmonic function cannot attain an interior maximum or minimum. These facts help explain the uniqueness of many equilibrium boundary-value problems. (web.stanford.edu)

Coordinate expressions

The Cartesian formula cannot be transferred unchanged to curvilinear coordinates: scale factors and volume elements must be included. In plane polar coordinates $(r,\theta)$,

In spherical coordinates $(r,\theta,\phi)$, with $\theta$ the polar angle,

The apparent singularities at the origin and poles arise from the coordinates rather than from the Euclidean operator itself. (ocw.mit.edu)

For a radial function $f(x)=F(r)$ in $\mathbb{R}^n$, where $r=|x|>0$,

This reduces rotationally symmetric Laplace and Poisson equations to ordinary differential equations. (math.stanford.edu)

Differential equations and physical applications

The Laplacian appears in several fundamental equations:

• Laplace’s equation: $\Delta u=0$, describing source-free equilibrium fields.
• Poisson’s equation: $\Delta u=f$, describing an equilibrium field with a prescribed source.
• Heat equation: $\partial_tu=\kappa\Delta u$, where $\kappa>0$, describing homogeneous isotropic diffusion.
• Wave equation: $\partial_t^2u=c^2\Delta u$, describing waves in a homogeneous isotropic medium. (math.mit.edu)

In electrostatics, the electric potential $V$ in a medium with constant permittivity satisfies

where $\rho$ is electric charge density. In a charge-free region this becomes Laplace’s equation. Analogous equations govern Newtonian gravitational potentials. (web.stanford.edu)

In nonrelativistic quantum mechanics, the Laplacian represents the spatial part of the kinetic-energy operator for a particle of mass $m$:

The Schrödinger equation therefore takes the form

Here $\psi$ is the wave function and $V$ is potential energy. (damtp.cam.ac.uk)

The ordinary Laplacian assumes an isotropic spatial response. When transport coefficients vary in space, or transport is direction-dependent, the relevant operator is generally

where $A(x)$ is a coefficient matrix, rather than a constant multiple of $\Delta$. (math.stanford.edu)

Boundary conditions and the energy identity

On a bounded domain $\Omega$, a differential equation involving the Laplacian generally requires boundary data. Common choices in a boundary-value problem include:

• Dirichlet conditions, prescribing $u$ on the boundary;
• Neumann conditions, prescribing its outward normal derivative $\partial_nu$;
• Robin conditions, prescribing a linear combination of the two. (arxiv.org)

The divergence theorem yields Green’s first identity. For sufficiently smooth real-valued $u,v$,

Taking $v=u$, with boundary conditions that eliminate the boundary term, gives

This connects the Laplacian to an energy functional measuring spatial variation. (math.stanford.edu)

As an operator on an Hilbert space such as $L^2(\Omega)$, the Laplacian is not specified by its differential expression alone: its domain and boundary conditions are essential. Suitable Dirichlet and Neumann realizations of $-\Delta$ are self-adjoint operators, but they have different spectra and kernels. On a bounded connected domain, constants form the Neumann kernel; homogeneous Dirichlet conditions exclude nonzero constants. (arxiv.org)

Fourier and spectral descriptions

With the Fourier transform convention using $e^{-ix\cdot\xi}$,

The Laplacian thus multiplies each spatial frequency by a factor proportional to the square of its magnitude. For the heat equation, the corresponding time-evolution factor is $e^{-\kappa t|\xi|^2}$, explaining why diffusion suppresses short-wavelength variations especially rapidly. (arxiv.org)

On a bounded smooth domain, standard Dirichlet or Neumann conditions lead to the eigenvalue problem

Its eigenfunctions can be chosen as an orthonormal basis of $L^2(\Omega)$. Under homogeneous boundary conditions, an initial heat distribution can be expanded into these modes:

The same spatial modes determine vibration frequencies through $\omega_k=c\sqrt{\lambda_k}$. (web.stanford.edu)

The spectral behavior depends on the underlying space. On all of $\mathbb{R}^n$, the standard operator $-\Delta$ has continuous spectrum $[0,\infty)$, rather than the discrete eigenvalue sequence characteristic of bounded domains. (arxiv.org)

Geometric generalization

On a manifold equipped with a Riemannian metric, the corresponding scalar operator is the Laplace–Beltrami operator:

In local coordinates,

where $|g|=\det(g_{ij})$ and $(g^{ij})$ is the inverse metric matrix. This is an intrinsic definition: its coordinate expressions describe the same geometrically defined operator. When the metric is Euclidean, it recovers the ordinary Laplacian. (math.mit.edu)

Laplacians also act on differential forms and other geometric objects. Such extensions require specifying the structure used to define the operator; a scalar Laplacian applied componentwise is not automatically an intrinsic operator on arbitrary tensor fields. (math.mit.edu)

Discrete and numerical forms

The finite difference method approximates the two-dimensional Laplacian on a square grid of spacing $h$ by the five-point stencil

This compares a grid value with its four nearest neighbors and converts a differential equation into a sparse system of linear equations. For a sufficiently smooth function, Taylor expansion shows that the interior approximation has error $O(h^2)$; the full numerical solution also depends on boundary treatment and stability. (math.mit.edu)

In graph theory, the graph Laplacian of an undirected weighted graph is

where $A$ is its symmetric adjacency matrix and $D$ contains the weighted vertex degrees. It acts on vertex values by

Its energy identity is

For positive edge weights, $L$ is positive semidefinite, and its zero eigenspace consists of functions constant on each connected component. This convention corresponds to the sign of $-\Delta$. Graph-Laplacian eigenvectors are used in spectral graph drawing and partitioning, extending the relationship between spatial variation, energy, and eigenmodes to discrete networks. (cs.yale.edu)

References

1. Lectures on PDEmath.stanford.edu
2. Math 220B Lecture Notesweb.stanford.edu
3. 155, Differential Analysis I, Fall 2021math.mit.edu
4. Lecture Notes for 18.155math.mit.edu
5. 303 Fall 2009math.mit.edu
6. Spectral Theory of Partial Differential Equations — Lecture Notesarxiv.org
7. Lecture 11: The Laplacian in Polar Coordinatesocw.mit.edu
8. MIT Mathematics: 18.085 Fall 2010math.mit.edu
9. Spectral Graph Theory, Lecture 2: The Laplaciancs.yale.edu
10. Spectral Graph Theory, Fall 2019: Syllabuscs.yale.edu