Laplace's Equation

Definition In three dimensions, the problem is to find twice-differentiable real-valued functions, of real variables, x, y, and z, such that

This is often written as

or

where div is the divergence, and grad is the gradient, or

where Δ is the Laplace operator. Solutions of Laplace's equation are called harmonic functions. If the right-hand side is specified as a given function, f(x, y, z), i.e.

then the equation is called "Poisson's equation." Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The partial differential operator, , or Δ, (which may be defined in any number of dimensions) is called the Laplace operator, or just the Laplacian.

[edit] Boundary conditions The Dirichlet problem for Laplace's equation consists in finding a solution on some domain D such that 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 and wait until the temperature in the interior 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 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. 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 homogenous differential equation), their sum (or any linear combination) is also a solution. This property, called the principle of superposition, is very useful, since solutions to complex problems can be constructed by summing simple solutions.

[edit] Laplace equation in two dimensions The Laplace equation in two independent variables has the form

[edit] Analytic functions The real and imaginary parts of a complex analytic function both satisfy the Laplace equation. That is, if z = x + iy, and if

then the necessary condition that f(z) be analytic is that the Cauchy-Riemann equations be satisfied:

It follows that

Therefore u satisfies the Laplace equation. A similar calculation shows that v also satisfies the Laplace equation. Conversely, given a harmonic function, it is the real part of an analytic function, f(z) (at least locally). If a trial form is

then the Cauchy-Riemann equations will be satisfied if we set

This relation does not determine ψ, but only its increments:

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

and thus ψ 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 r and θ are polar coordinates and

then a corresponding analytic function is

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

with suitably defined coefficients whose real and imaginary parts are given by

Therefore

which is a Fourier series for f.

[edit] Fluid flow 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 condition that the flow be incompressible is that

and the condition that the flow be irrotational is that

If we define the differential of a function ψ by

then the incompressibility 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 ψ are given by

and the irrotationality condition implies that ψ satisfies the Laplace equation. The harmonic function φ that is conjugate to ψ is called the velocity potential. The CauchyRiemann equations imply that

Thus every analytic function corresponds to a steady incompressible, irrotational fluid flow in the plane. The real part is the velocity potential, and the imaginary part is the stream function.

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

and

where ρ is the charge density. The first Maxwell equation is the integrability condition for the differential

so the electric potential φ may be constructed to satisfy

The second of Maxwell's equations then implies that

which is the Poisson equation. It is important to note that the Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.

[edit] Laplace equation in three dimensions [edit] Fundamental solution A fundamental solution of Laplace's equation satisfies

where the Dirac delta function δ denotes a unit source concentrated at the point No function has this property, 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). 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

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

on a sphere of radius r that is centered around the source point, and hence

A similar argument shows that in two dimensions

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

may satisfy

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 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

where R denotes the distance to the source point P and R' denotes the distance to the reflected point P'. A consequence of this expression for the Green's function is the Poisson integral formula. 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 inside the sphere is given by

where

A simple consequence of this formula is that if u is a harmonic function, then the value of u 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.

LaPlace's and Poisson's Equations A useful approach to the calculation of electric potentials is to relate that potential to the charge density which gives rise to it. The electric field is related to the charge density by the divergence relationship

and the electric field is related to the electric potential by a gradient relationship

Therefore the potential is related to the charge density by Poisson's equation

Index Electric field concepts

In a charge-free region of space, this becomes LaPlace's equation

This mathematical operation, the divergence of the gradient of a function, is called the LaPlacian. Expressing the LaPlacian in different coordinate systems to take advantage of the symmetry of a charge distribution helps in the solution for the electric potential V. For example, if the charge distribution has spherical symmetry, you use the LaPlacian in spherical polar coordinates. Since the potential is a scalar function, this approach has advantages over trying to calculate the electric field directly. Once the potential has been calculated, the electric field can be computed by taking the gradient of the potential. Example: sphere of uniform charge HyperPhysics***** Electricity and Magnetism

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Potential of a Uniform Sphere of Charge The use of Poisson's and Laplace's equations will be explored for a uniform sphere of charge. In spherical polar coordinates, Poisson's equation takes the form:

but since there is full spherical symmetry here, the derivatives with respect to θ and φ must be zero, leaving the form

Examining first the region outside the sphere, Laplace's law applies.

Since the zero of potential is arbitrary, it is reasonable to choose the zero of potential at infinity, the standard practice with localized charges. This gives the value b=0. Since the sphere of charge will look like a point charge at large distances, we may conclude that

so the solution to LaPlace's law outside the sphere is

Now examining the potential inside the sphere, the potential must have a term of order r2 to give a constant on the left side of the equation, so the solution is of the form

Substituting into Poisson's equation gives

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