Laplace's Equation · aiwiki.page

Laplace’s equation is the partial differential equation

\Delta u=0,

where $\Delta$ is the Laplace operator and $u$ is an unknown scalar function. Its solutions are called harmonic functions. The equation describes many source-free equilibrium phenomena, including steady heat conduction and electric or gravitational potentials away from their sources. It is also a central equation in mathematical analysis. (damtp.cam.ac.uk)

Mathematical formulation

For a twice continuously differentiable function $u$ on an open set $\Omega$ in $n$ -dimensional Euclidean space, Laplace’s equation in Cartesian coordinates is

\Delta u =\sum_{j=1}^{n}\frac{\partial^2u}{\partial x_j^2} =0.

In two dimensions this becomes $u_{xx}+u_{yy}=0$ ; in three dimensions it becomes $u_{xx}+u_{yy}+u_{zz}=0$ . The subscripts denote partial derivatives. Replacing the zero on the right by a prescribed source function gives Poisson’s equation, with either sign convention commonly used. (damtp.cam.ac.uk)

Laplace’s equation is linear and homogeneous: any linear combination of solutions is again a solution. It is the basic example of an elliptic partial differential equation. Unlike a time-evolution equation, it generally determines a spatial field through boundary conditions rather than initial conditions. The superposition principle applies to the equation, although a particular boundary condition need not be preserved by every combination of solutions. (damtp.cam.ac.uk)

Physical interpretation and applications

In electrostatics, the electric field is related to the electric potential $V$ by

\mathbf E=-\nabla V,

where $\nabla V$ is its gradient. In a medium of constant permittivity, Gauss’s law gives

\Delta V=-\frac{\rho_e}{\varepsilon}.

Thus $V$ satisfies Laplace’s equation wherever the electric charge density $\rho_e$ vanishes. “Source-free” is a local condition: charges outside the region can still produce a nonzero field inside it. (damtp.cam.ac.uk)

For Newtonian gravity, the gravitational potential satisfies

\Delta\Phi=4\pi G\rho_m,

so it is harmonic outside the mass distribution. For steady heat conduction in a homogeneous, isotropic material without internal heat sources, the temperature satisfies $\Delta T=0$ . This follows by setting the time derivative to zero in the heat equation. (damtp.cam.ac.uk)

Another application is incompressible, irrotational fluid flow. Where a velocity potential exists, the velocity can be written as $\mathbf v=\nabla\phi$ ; incompressibility then gives $\nabla\cdot\mathbf v=\Delta\phi=0$ . (physics.uoguelph.ca)

Boundary-value problems

The equation alone does not select a unique solution. A boundary-value problem specifies additional information on the boundary $\partial\Omega$ . Two principal choices are:

Dirichlet conditions: prescribe $u=g$ on $\partial\Omega$ .
Neumann conditions: prescribe the outward normal derivative $\partial u/\partial n=h$ on $\partial\Omega$ .

For a bounded domain, a harmonic function continuous on its closure is uniquely determined by its Dirichlet boundary values. For a connected domain, a solvable pure Neumann problem determines the solution only up to an additive constant. (ocw.mit.edu)

For Laplace’s equation, Neumann data must satisfy

\int_{\partial\Omega}h\,dS=0.

The divergence theorem explains this compatibility condition: the total outward flux equals $\int_\Omega\Delta u\,dx$ , which is zero. Boundary data violating it admit no solution. (ocw.mit.edu)

Exterior problems concern the region outside an object. They also require an appropriate condition at infinity, such as a prescribed limiting value or decay, to distinguish the intended solution. (math.mit.edu)

Properties of harmonic functions

The mean-value property states that a harmonic function’s value at the center of any ball contained in its domain equals its average over that ball and over its boundary sphere:

u(x)=\frac{1}{|\partial B_r(x)|} \int_{\partial B_r(x)}u(y)\,dS_y.

The averaging is exact for every admissible radius, not merely an approximation for small neighborhoods. (web.stanford.edu)

The maximum principle states that a harmonic function continuous on the closure of a bounded domain attains its maximum and minimum on the boundary. The strong version says that attaining a maximum or minimum over a connected domain at an interior point forces the function to be constant. This yields uniqueness for the Dirichlet problem: the difference of two solutions with identical boundary values is harmonic with zero boundary values and must vanish. (ocw.mit.edu)

Harmonic functions are infinitely differentiable and, more strongly, locally real analytic. Their interior smoothness therefore exceeds the twice-continuous differentiability required in the classical definition. This does not imply comparable smoothness at a rough boundary. (web.stanford.edu)

Representative solutions and analytical methods

A radially symmetric solution $u(x)=U(r)$ , where $r=|x|$ , satisfies

U''(r)+\frac{n-1}{r}U'(r)=0.

Away from the origin, its general form is

U(r)= \begin{cases} A+B\log r,&n=2,\\ A+Br^{2-n},&n\geq3. \end{cases}

In three dimensions the nonconstant term is proportional to $1/r$ . These functions are harmonic away from their singularity; they are not source-free solutions at the origin. With suitable normalization, they provide fundamental solutions for the Laplace operator. (ocw.mit.edu)

A Green’s function incorporates the response to a point source and can be adapted to a domain and its boundary conditions. Single-layer and double-layer potentials represent solutions through integrals over the boundary, converting a differential equation in the domain into an integral equation on its surface. (math.mit.edu)

For simple geometries, separation of variables constructs solutions from products of functions of individual coordinates. Expansions in Fourier series allow boundary data to be matched mode by mode. For example, in the unit disk, the boundary mode $\cos(m\theta)$ has the regular harmonic extension $r^m\cos(m\theta)$ , for a positive integer $m$ . (damtp.cam.ac.uk)

Numerical approximation

The finite difference method replaces derivatives by differences between grid values. On a square grid of spacing $h$ , the standard five-point approximation is

\frac{ u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{i,j} }{h^2}=0.

Thus each interior grid value equals the arithmetic average of its four nearest neighbors. This is a discrete counterpart of harmonic averaging. (cfm.brown.edu)

Prescribed boundary values turn these equations into a system of linear equations for the unknown interior values. For sufficiently smooth solutions, the stencil has second-order local truncation error, $O(h^2)$ . Actual accuracy also depends on how the boundary is represented and how accurately the resulting algebraic system is solved. (cfm.brown.edu)

Historical development and scope

The equation is named after Pierre-Simon Laplace, who employed it in gravitational potential theory and celestial mechanics. It appears in his Traité de mécanique céleste, whose first two volumes were published in 1799. Despite its modern name, the equation was known before Laplace’s work. (mathshistory.st-andrews.ac.uk)

The distinction between Laplace’s equation and more general equilibrium equations is important. Internal sources produce Poisson’s equation. Spatially varying transport properties generally lead to a divergence-form equation such as

\nabla\cdot\bigl(k(x)\nabla u\bigr)=0,

rather than $\Delta u=0$ . Laplace’s equation follows from this form when $k$ is a nonzero constant. Similarly, singular source locations must be excluded when treating their potentials as harmonic functions. (damtp.cam.ac.uk)

References

Four Important Linear PDEocw.mit.edu
Laplace’s Equationweb.stanford.edu
Fast Algorithms for Integral Equationsmath.mit.edu
Chapter 10: Laplace’s Equationphysics.uoguelph.ca
Numerical Solutions of Laplace Equationcfm.brown.edu
Fourier Series Solution of Laplace’s Equationmitocw.ups.edu.ec
Pierre-Simon Laplace — MacTutor History of Mathematicsmathshistory.st-andrews.ac.uk