In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as or where is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, . Its use to represent the Laplacian should not be confused with this use. is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function.
If the right-hand side is specified as a given function, , we have
This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. Laplace's equation is also a special case of the Helmholtz equation.
The general theory of solutions to Laplace's equation is known as potential theory. The twice continuously differentiable solutions of Laplace's equation are the harmonic functions,Stewart, James. Calculus : Early Transcendentals. 7th ed., Brooks/Cole, Cengage Learning, 2012. Chapter 14: Partial Derivatives. p. 908. . which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics. In the study of heat conduction, the Laplace equation is the steady-state heat equation.Zill, Dennis G, and Michael R Cullen. Differential Equations with Boundary-Value Problems. 8th edition / ed., Brooks/Cole, Cengage Learning, 2013. Chapter 12: Boundary-value Problems in Rectangular Coordinates. p. 462. . In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.
Forms in different coordinate systems In rectangular coordinates,Griffiths, David J. Introduction to Electrodynamics. 4th ed., Pearson, 2013. Inner front cover. .
In cylindrical coordinates,
In spherical coordinates, using the convention,
More generally, in arbitrary curvilinear coordinates ,
or
where is the Euclidean metric tensor relative to the new coordinates and denotes its Christoffel symbols.
Boundary conditions
The Dirichlet problem for Laplace's equation consists of finding a solution on some domain such that on the boundary of is equal to some given function. Since the Laplace operator appears in the heat equation, one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.
The Neumann boundary conditions for Laplace's equation specify not the function itself on the boundary of but its normal derivative. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of is zero.
Solutions of Laplace's equation are called harmonic functions; they are all analytic within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful. For example, solutions to complex problems can be constructed by summing simple solutions.
Details
For a domain , the most common boundary value problems for Laplace's equation are the Dirichlet problem, in which the boundary values of the unknown function are prescribed, and the Neumann problem, in which its outward normal derivative is prescribed.
If is bounded and connected, then a harmonic function is uniquely determined by its Dirichlet boundary values; this follows from the maximum principle. For the Neumann problem, uniqueness holds only up to an additive constant. Moreover, if is harmonic in , then the divergence theorem implies the compatibility condition
Hence a necessary condition for solvability of the Neumann problem with prescribed flux is A third classical boundary condition is the Robin boundary condition, which prescribes a linear combination of the function and its normal derivative on the boundary.
In special domains, solutions can be written explicitly by integral formulas. For the unit disk , the solution of the Dirichlet problem with continuous boundary data is given by the Poisson kernel formula
Similarly, in the upper half-space , harmonic functions with suitable boundary data are represented by the Poisson integral
These kernels are densities of the harmonic measure with respect to boundary measure in these model domains.
Existence theory and boundary regularity
A classical approach to the Dirichlet problem for Laplace's equation is the Perron method, which constructs a candidate solution as the supremum of all subharmonic functions lying below the prescribed boundary data. The resulting Perron solution is harmonic in the domain; the subtle question is whether it attains the desired boundary values at every boundary point.
This leads to the notion of a regular boundary point. A boundary point is regular if the solution of the Dirichlet problem converges to the prescribed boundary value there. A sufficient condition is the existence of a barrier at that point, namely a superharmonic function that vanishes at the point and is positive elsewhere near the boundary.
For Laplace's equation, regularity of boundary points is characterized by the classical Wiener criterion, which gives a necessary and sufficient condition in terms of capacity. In this way, solvability of the Dirichlet problem is tied to the fine geometric structure of the boundary.
Weak solutions and Dirichlet principle
Laplace's equation can also be interpreted in a weak sense. A function is called weakly harmonic if
for every test function . By Weyl's lemma, every weakly harmonic function is in fact smooth, and indeed real analytic.
Harmonic functions also admit a variational characterization. Among functions with fixed boundary values, the solutions of Laplace's equation are exactly the minimizers of the Dirichlet energy
This is known as Dirichlet's principle.
Kelvin transform
A classical symmetry of Laplace's equation is inversion in the unit sphere. If is harmonic in a domain , then its Kelvin transform
is harmonic in the inverted domain
In dimension , the factor is equal to , so the transform reduces to composition with inversion.
The Kelvin transform is useful for converting interior problems into exterior problems, for studying isolated singularities, and for analyzing the behavior of harmonic functions at infinity.
In two dimensions Laplace's equation in two independent variables in rectangular coordinates has the form
Analytic functions
The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if , and if
then the necessary condition that be analytic is that and be differentiable and that the Cauchy–Riemann equations be satisfied:
where is the first partial derivative of with respect to . It follows that
Therefore satisfies the Laplace equation. A similar calculation shows that also satisfies the Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, (at least locally). If a trial form is
then the Cauchy–Riemann equations will be satisfied if we set
This relation does not determine , but only its increments:
The Laplace equation for implies that the integrability condition for is satisfied:
and thus may be defined by a line integral. The integrability condition and Stokes' theorem implies that the value of the line integral connecting two points is independent of the path. The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions. This construction is only valid locally, or provided that the path does not loop around a singularity. For example, if and are polar coordinates and
then a corresponding analytic function is
However, the angle is single-valued only in a region that does not enclose the origin.
The close connection between the Laplace equation and analytic functions implies that any solution of the Laplace equation has derivatives of all orders, and can be expanded in a power series, at least inside a circle that does not enclose a singularity. This is in sharp contrast to solutions of the wave equation, which generally have less regularity.
There is an intimate connection between power series and Fourier series. If we expand a function in a power series inside a circle of radius , this means that
with suitably defined coefficients whose real and imaginary parts are given by
Therefore
which is a Fourier series for . These trigonometric functions can themselves be expanded, using multiple angle formulae.
Fluid flow
Let the quantities and be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions. The continuity condition for an incompressible flow is that
and the condition that the flow be irrotational is that
If we define the differential of a function by
then the continuity condition is the integrability condition for this differential: the resulting function is called the stream function because it is constant along flow lines. The first derivatives of are given by
and the irrotationality condition implies that satisfies the Laplace equation. The harmonic function that is conjugate to is called the velocity potential. The Cauchy–Riemann equations imply that
Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.
Electrostatics According to Maxwell's equations, an electric field in two space dimensions that is independent of time satisfies
and
where is the charge density. The first Maxwell equation is the integrability condition for the differential
so the electric potential may be constructed to satisfy
The second of Maxwell's equations then implies that
which is the Poisson equation. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.
In three dimensions
Fundamental solution A fundamental solution of Laplace's equation satisfies
where the Dirac delta function denotes a unit source concentrated at the point . No function has this property: in fact it is a distribution rather than a function; but it can be thought of as a limit of functions whose integrals over space are unity, and whose support (the region where the function is non-zero) shrinks to a point (see weak solution). It is common to take a different sign convention for this equation than one typically does when defining fundamental solutions. This choice of sign is often convenient to work with because −Δ is a positive operator. The definition of the fundamental solution thus implies that, if the Laplacian of is integrated over any volume that encloses the source point, then
The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance from the source point. If we choose the volume to be a ball of radius around the source point, then Gauss's divergence theorem implies that
It follows that
on a sphere of radius that is centered on the source point, and hence