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 systemsEdit
More generally, in curvilinear coordinates,
The Dirichlet problem for Laplace's equation consists of finding a solution φ on some domain D such that φ on the boundary of D 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 doesn't 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 φ itself on the boundary of D 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 D 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 φ 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.
In two dimensionsEdit
Laplace's equation in two independent variables in rectangular coordinates has the form
The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z = x + iy, and if
However, the angle θ 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 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. The continuity condition for an incompressible flow is that
According to Maxwell's equations, an electric field (u, v) in two space dimensions that is independent of time satisfies
In three dimensionsEdit
A fundamental solution of Laplace's equation satisfies
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 r from the source point. If we choose the volume to be a ball of radius a around the source point, then Gauss' divergence theorem implies that
It follows 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. A similar argument shows that in two dimensions
A Green's function is a fundamental solution that also satisfies a suitable condition on the boundary S of a volume V. For instance,
Now if u is any solution of the Poisson equation in V:
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
Note that if P is inside the sphere, then P′ will be outside the sphere. The Green's function is then given by
Laplace's spherical harmonicsEdit
Consider the problem of finding solutions of the form f(r, θ, φ) = R(r) Y(θ, φ). By separation of variables, two differential equations result by imposing Laplace's equation:
The second equation can be simplified under the assumption that Y has the form Y(θ, φ) = Θ(θ) Φ(φ). Applying separation of variables again to the second equation gives way to the pair of differential equations
for some number m. A priori, m is a complex constant, but because Φ must be a periodic function whose period evenly divides 2π, m is necessarily an integer and Φ is a linear combination of the complex exponentials e±imφ. The solution function Y(θ, φ) is regular at the poles of the sphere, where θ = 0, π. Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm–Liouville problem that forces the parameter λ to be of the form λ = ℓ (ℓ + 1) for some non-negative integer with ℓ ≥ |m|; this is also explained below in terms of the orbital angular momentum. Furthermore, a change of variables t = cos θ transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pℓm(cos θ) . Finally, the equation for R has solutions of the form R(r) = A rℓ + B r−ℓ − 1; requiring the solution to be regular throughout R3 forces B = 0.
Here the solution was assumed to have the special form Y(θ, φ) = Θ(θ) Φ(φ). For a given value of ℓ, there are 2ℓ + 1 independent solutions of this form, one for each integer m with −ℓ ≤ m ≤ ℓ. 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. In particular, the colatitude θ, or polar angle, ranges from 0 at the North Pole, to π/2 at the Equator, to π at the South Pole, and the longitude φ, or azimuth, may assume all values with 0 ≤ φ < 2π. For a fixed integer ℓ, every solution Y(θ, φ) of the eigenvalue problem
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,
For , the solid harmonics with negative powers of are chosen instead. In that case, one needs to expand the solution of known regions in Laurent series (about ), instead of Taylor series (about ), to match the terms and find .
Now, the electric field can be expressed as the negative gradient of the electric potential ,
Plugging this relation into Gauss's law, we obtain Poisson's equation for electricity,
In the particular case of a source-free region, and Poisson's equation reduces to Laplace's equation for the electric potential.
If the electrostatic potential is specified on the boundary of a region , then it is uniquely determined. If is surrounded by a conducting material with a specified charge density , and if the total charge is known, then is also unique.
A potential that doesn't satisfy Laplace's equation together with the boundary condition is an invalid electrostatic potential.
Let be the gravitational field, the mass density, and the gravitational constant. Then Gauss's law for gravitation in differential form is
The gravitational field is conservative and can therefore be expressed as the negative gradient of the gravitational potential:
Using the differential form of Gauss's law of gravitation, we have
In empty space, and we have
In the Schwarzschild metricEdit
- 6-sphere coordinates, a coordinate system under which Laplace's equation becomes R-separable
- Helmholtz equation, a general case of Laplace's equation.
- Spherical harmonic
- Quadrature domains
- Potential theory
- Potential flow
- Bateman transform
- Earnshaw's theorem uses the Laplace equation to show that stable static ferromagnetic suspension is impossible
- Vector Laplacian
- Fundamental solution
- 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.
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- The approach to spherical harmonics taken here is found in (Courant & Hilbert 1966, §V.8, §VII.5) harv error: no target: CITEREFCourantHilbert1966 (help).
- Physical applications often take the solution that vanishes at infinity, making A = 0. This does not affect the angular portion of the spherical harmonics.
- Griffiths, David J. Introduction to Electrodynamics. Fourth ed., Pearson, 2013. Chapter 2: Electrostatics. p. 83-4. ISBN 978-1-108-42041-9.
- Griffiths, David J. Introduction to Electrodynamics. Fourth ed., Pearson, 2013. Chapter 3: Potentials. p. 119-121. ISBN 978-1-108-42041-9.
- Persides, S. (1973). "The Laplace and poisson equations in Schwarzschild's space-time". Journal of Mathematical Analysis and Applications. 43 (3): 571–578. Bibcode:1973JMAA...43..571P. doi:10.1016/0022-247X(73)90277-1.
- Evans, L. C. (1998). Partial Differential Equations. Providence: American Mathematical Society. ISBN 978-0-8218-0772-9.
- Petrovsky, I. G. (1967). Partial Differential Equations. Philadelphia: W. B. Saunders.
- Polyanin, A. D. (2002). Handbook of Linear Partial Differential Equations for Engineers and Scientists. Boca Raton: Chapman & Hall/CRC Press. ISBN 978-1-58488-299-2.
- Sommerfeld, A. (1949). Partial Differential Equations in Physics. New York: Academic Press.
- Zachmanoglou, E. C. (1986). Introduction to Partial Differential Equations with Applications. New York: Dover.
- "Laplace equation", Encyclopedia of Mathematics, EMS Press, 2001 
- Laplace Equation (particular solutions and boundary value problems) at EqWorld: The World of Mathematical Equations.
- Example initial-boundary value problems using Laplace's equation from exampleproblems.com.
- Weisstein, Eric W. "Laplace's Equation". MathWorld.
- Find out how boundary value problems governed by Laplace's equation may be solved numerically by boundary element method