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

<math display="block"> \nabla^2\! f = 0 </math> or <math display="block"> \Delta f = 0,</math>

where <math> \Delta = \nabla \cdot \nabla = \nabla^2</math> is the Laplace operator, <math>\nabla \cdot</math> is the divergence operator (also symbolized "div"), <math>\nabla</math> is the gradient operator (also symbolized "grad"), and <math>f (x, y, z)</math> 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, <math>h(x, y, z)</math>, we have

<math display="block">\Delta f = h</math>

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,

<math display="block"> \nabla^2 f = \frac{\partial^2 f}{\partial x^2 } + \frac{\partial^2 f}{\partial y^2 } + \frac{\partial^2 f}{\partial z^2 } = 0.</math>

In cylindrical coordinates,

If <math>D</math> is bounded and connected, then a harmonic function is uniquely determined by its Dirichlet boundary values; this follows from the maximum principle. 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 <math>u\in H^1_{\mathrm{loc(\Omega)</math> is called weakly harmonic if

<math display="block">\int_\Omega \nabla u\cdot \nabla \varphi\,dx=0</math>

for every test function <math>\varphi\in C_c^\infty(\Omega)</math>. By Weyl's lemma, every weakly harmonic function is in fact smooth, and indeed real analytic.

<math display="block"> \nabla^2 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} = 0.</math>

Consider the problem of finding solutions of the form . By separation of variables, two differential equations result by imposing Laplace's equation:

<math display="block">\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) = \lambda,\qquad \frac{1}{Y}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial Y}{\partial\theta}\right) + \frac{1}{Y}\frac{1}{\sin^2\theta}\frac{\partial^2Y}{\partial\varphi^2} = -\lambda.</math>

The second equation can be simplified under the assumption that has the form . Applying separation of variables again to the second equation gives way to the pair of differential equations

<math display="block">\frac{1}{\Phi} \frac{d^2 \Phi}{d\varphi^2} = -m^2</math>

<math display="block">\lambda\sin^2\theta + \frac{\sin\theta}{\Theta} \frac{d}{d\theta} \left(\sin\theta \frac{d\Theta}{d\theta}\right) = m^2</math>

for some number . A priori, is a complex constant, but because must be a periodic function whose period evenly divides , is necessarily an integer and is a linear combination of the complex exponentials . The solution function is regular at the poles of the sphere, where . Imposing this regularity in the solution of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter to be of the form for some non-negative integer with ; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial . Finally, the equation for has solutions of the form ; requiring the solution to be regular throughout forces .

Here the solution was assumed to have the special form . For a given value of , there are independent solutions of this form, one for each integer with . These angular solutions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials:

<math display="block"> Y_\ell^m (\theta, \varphi ) = N e^{i m \varphi } P_\ell^m (\cos{\theta} )</math>

which fulfill

<math display="block"> r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ).</math>

Here is called a spherical harmonic function of degree and order , is an associated Legendre polynomial, is a normalization constant, and and represent colatitude and longitude, respectively. In particular, the colatitude , or polar angle, ranges from at the North Pole, to at the Equator, to at the South Pole, and the longitude , or azimuth, may assume all values with . For a fixed integer , every solution of the eigenvalue problem

<math display="block"> r^2\nabla^2 Y = -\ell (\ell + 1 ) Y</math>

is a linear combination of . In fact, for any such solution, is the expression in spherical coordinates of a homogeneous polynomial that is harmonic (see below), and so counting dimensions shows that there are linearly independent such polynomials.

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 ,

<math display="block"> f(r, \theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m r^\ell Y_\ell^m (\theta, \varphi ), </math>

where the are constants and the factors are known as solid harmonics. Such an expansion is valid in the ball

For <math> r > R</math>, the solid harmonics with negative powers of <math>r</math> are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about <math>r=\infty</math>), instead of Taylor series (about <math>r = 0</math>), to match the terms and find <math>f^m_\ell</math>.

Electrostatics and magnetostatics

Let <math>\mathbf{E}</math> be the electric field, <math>\rho</math> be the electric charge density, and <math>\varepsilon_0</math> be the permittivity of free space. Then Gauss's law for electricity (Maxwell's first equation) in differential form states

<math display="block">\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}.</math>

Now, the electric field can be expressed as the negative gradient of the electric potential <math>V</math>,

<math display="block">\mathbf E=-\nabla V,</math>

if the field is irrotational, <math>\nabla \times \mathbf{E} = \mathbf{0}</math>. The irrotationality of <math>\mathbf{E}</math> is also known as the electrostatic condition.

For the magnetic field, when there is no free current, <math display="block">\nabla\times\mathbf{H} = \mathbf{0}.</math>We can thus define a magnetic scalar potential, , as

<math display="block">\mathbf{H} = -\nabla\psi.</math>With the definition of :

<math display="block">\nabla\cdot\mathbf{B} = \mu_{0}\nabla\cdot\left(\mathbf{H} + \mathbf{M}\right) = 0,</math>

it follows that

<math display="block">\nabla^2 \psi = -\nabla\cdot\mathbf{H} = \nabla\cdot\mathbf{M}.</math>

Similar to electrostatics, in a source-free region, <math>\mathbf{M} = 0</math> and Poisson's equation reduces to Laplace's equation for the magnetic scalar potential ,

<math display="block">\nabla^2 \psi = 0</math>

A potential that does not satisfy Laplace's equation together with the boundary condition is an invalid electrostatic or magnetic scalar potential.

Gravitation

Let <math>\mathbf{g}</math> be the gravitational field, <math>\rho</math> the mass density, and <math>G</math> the gravitational constant. Then Gauss's law for gravitation in differential form is

<math display="block">\nabla\cdot\mathbf g=-4\pi G\rho.</math>

The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential:

<math display="block">\begin{align}

\mathbf g &= -\nabla V, \\

\nabla\cdot\mathbf g &= \nabla\cdot(-\nabla V) = -\nabla^2 V, \\

\implies\nabla^2 V &= -\nabla\cdot\mathbf g.

\end{align}</math>

Using the differential form of Gauss's law of gravitation, we have

<math display="block">\nabla^2 V = 4\pi G\rho,</math>

which is Poisson's equation for gravitational fields.

The distribution of the exit point <math>B_{\tau_D}</math> on <math>\partial D</math> is called the harmonic measure of <math>D</math>. Thus harmonic measure gives a probabilistic solution of the Dirichlet problem: the harmonic extension of a boundary function is obtained by averaging the boundary values against the exit distribution of Brownian motion.

This probabilistic viewpoint also gives short proofs of several basic properties of harmonic functions, including the mean value property, the maximum principle, and uniqueness for the Dirichlet problem.

Harmonic measure

For a bounded domain <math>D\subset \mathbf R^n</math> and a point <math>x\in D</math>, the solution of the Dirichlet problem with boundary data <math>f</math> can be represented in the form

<math display="block">u(x)=\int_{\partial D} f(\xi)\,d\omega_D^x(\xi),</math>

where <math>\omega_D^x</math> is a probability measure on <math>\partial D</math> called the harmonic measure of <math>D</math> with pole at <math>x</math>. In this way, harmonic measure encodes the influence of the boundary values on the interior solution.

In classical domains such as the disk or ball, harmonic measure is absolutely continuous with respect to surface measure, and its density is the Poisson kernel. Probabilistically, the harmonic measure is the distribution of the point where Brownian motion started at <math>x</math> first exits the domain.

In the Schwarzschild metric

S. Persides solved the Laplace equation in Schwarzschild spacetime on hypersurfaces of constant . Using the canonical variables , , the solution is

<math display="block">\Psi(r,\theta,\varphi) = R(r)Y_l(\theta,\varphi),</math>

where is a spherical harmonic function, and

R(r) = (-1)^l\frac{(l!)^2r_s^l}{(2l)!}P_l\left(1-\frac{2r}{r_s}\right)+(-1)^{l+1}\frac{2(2l+1)!}{(l)!^2r_s^{l+1Q_l\left(1-\frac{2r}{r_s}\right).

</math>

Here and are Legendre functions of the first and second kind, respectively, while is the Schwarzschild radius. The parameter is an arbitrary non-negative integer.