Laplace's equation

The twice continuously differentiable solutions of Laplace's equation are the harmonic functions,[1] which are important in multiple branches of physics, notably electrostatics, gravitation, and fluid dynamics.

[2] In general, Laplace's equation describes situations of equilibrium, or those that do not depend explicitly on time.

where gij is the Euclidean metric tensor relative to the new coordinates and Γ denotes its Christoffel symbols.

Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore.

Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of D alone.

The real and imaginary parts of a complex analytic function both satisfy the Laplace equation.

The Laplace equation for φ implies that the integrability condition for ψ is satisfied:

The resulting pair of solutions of the Laplace equation are called conjugate harmonic functions.

This is in sharp contrast to solutions of the wave equation, which generally have less regularity[citation needed].

If we expand a function f in a power series inside a circle of radius R, this means that

Let the quantities u and v be the horizontal and vertical components of the velocity field of a steady incompressible, irrotational flow in two dimensions.

Thus every analytic function corresponds to a steady incompressible, irrotational, inviscid fluid flow in the plane.

According to Maxwell's equations, an electric field (u, v) in two space dimensions that is independent of time satisfies

where the Dirac delta function δ denotes a unit source concentrated at the point (x′, y′, z′).

The definition of the fundamental solution thus implies that, if the Laplacian of u is integrated over any volume that encloses the source point, then

If we choose the volume to be a ball of radius a around the source point, then Gauss's divergence theorem implies that

Note that, with the opposite sign convention (used in physics), this is the potential generated by a point particle, for an inverse-square law force, arising in the solution of Poisson equation.

Note that, with the opposite sign convention, this is the potential generated by a pointlike sink (see point particle), which is the solution of the Euler equations in two-dimensional incompressible flow.

A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. For instance,

and u assumes the boundary values g on S, then we may apply Green's identity, (a consequence of the divergence theorem) which states that

The notations un and Gn denote normal derivatives on S. In view of the conditions satisfied by u and G, this result simplifies to

Thus the Green's function describes the influence at (x′, y′, z′) of the data f and g. For the case of the interior of a sphere of radius a, the Green's function may be obtained by means of a reflection (Sommerfeld 1949): the source point P at distance ρ from the center of the sphere is reflected along its radial line to a point P' that is at a distance

Let ρ, θ, and φ be spherical coordinates for the source point P. Here θ 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 g inside the sphere is given by (Zachmanoglou & Thoe 1986, p. 228)

These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

Here Yℓm is called a spherical harmonic function of degree ℓ and order m, Pℓm is an associated Legendre polynomial, N is a normalization constant, and θ and φ represent colatitude and longitude, respectively.

The general solution to Laplace's equation in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor rℓ,

Then Gauss's law for electricity (Maxwell's first equation) in differential form states[5]

S. Persides[8] solved the Laplace equation in Schwarzschild spacetime on hypersurfaces of constant t. Using the canonical variables r, θ, φ the solution is

Here Pl and Ql are Legendre functions of the first and second kind, respectively, while rs is the Schwarzschild radius.