Laplace's equation

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, is the divergence operator, is the gradient operator, 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, 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. 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,
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, their sum 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.

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, . 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 shrinks to a point. 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
Note that, with the opposite sign convention, this is the potential generated by a point particle, for an inverse-square law force, arising in the solution of the Poisson equation. A similar argument shows that in two dimensions
where denotes the natural logarithm. Note that, with the opposite sign convention, this is the potential generated by a pointlike sink, which is the solution of the Euler equations in two-dimensional incompressible flow.

Green's function

A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary of a volume. For instance,
may satisfy
Now if is any solution of the Poisson equation in :
and assumes the boundary values on, then we may apply Green's identity, which states that
The notations u_n and G_n denote normal derivatives on. In view of the conditions satisfied by and, this result simplifies to
Thus the Green's function describes the influence at of the data and. For the case of the interior of a sphere of radius, the Green's function may be obtained by means of a reflection : the source point at distance from the center of the sphere is reflected along its radial line to a point P that is at a distance
Note that if is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by
where denotes the distance to the source point and denotes the distance to the reflected point P''′. A consequence of this expression for the Green's function is the Poisson integral formula'''. Let,, and be spherical coordinates for the source point. Here denotes the angle with the vertical axis, which is contrary to the usual American mathematical notation, but agrees with standard European and physical practice. Then the solution of the Laplace equation with Dirichlet boundary values inside the sphere is given by
where
is the cosine of the angle between and. A simple consequence of this formula is that if is a harmonic function, then the value of at the center of the sphere is the mean value of its values on the sphere. This mean value property immediately implies that a non-constant harmonic function cannot assume its maximum value at an interior point.