The Cluster's Dirichlet Problem

A page about the Dirichlet problem — finding a function satisfying Laplace's equation with specified boundary values.

The Dirichlet problem

The Dirichlet problem asks: given a region V and a boundary surface S, find a function f such that: nabla^2 f = 0 in V f = g on S where g is a given function on the boundary. In electrostatics: find the potential V(r) inside a volume with fixed potentials on the boundary.

Uniqueness theorem: if f_1 and f_2 both satisfy the Dirichlet problem, then f_1 = f_2 everywhere in V. This follows from Green's first identity and the fact that the solution to Laplace's equation with fixed boundary values is unique.

In the cluster, the edit Dirichlet problem asks for an edit function satisfying an edit Laplace equation.

The Green's function solution

The solution to the Dirichlet problem is given by the Green's function for the region: f(x) = - (1 / (4 pi)) integral_S g(x') (partial G / partial n') dA' where G(x, x') is the Dirichlet Green's function: nabla'^2 G(x, x') = -4 pi delta(x - x') G(x, x') = 0 for x' on S (boundary condition)

The Green's function encodes the geometry of the region. For free space: G(x, x') = 1 / |x - x'|.

In the cluster, the edit Green's function solution gives an edit geometry encoding.

The method of images

For certain geometries, the Green's function can be found by the method of images. For a conducting sphere of radius R centered at the origin with a point charge at distance a from the center: Place an image charge q' = -q R / a at distance b = R^2 / a from the center. The potential on the sphere surface is zero.

For a conducting plane at z = 0: Place an image charge q' = -q at -z_0. The potential is zero on the plane.

The method of images gives the exact solution for:

• Infinite plane
• Sphere (with grounded or fixed potential)
• Sphere with a fixed charge (not grounded)
• Two intersecting conducting planes at angles pi / n

In the cluster, the edit method of images gives an edit exact solution.

The Dirichlet-to-Neumann map

The Dirichlet-to-Neumann map relates the boundary values g(x) to the normal derivative (partial f / partial n) on the boundary. This map is fundamental in:

• Inverse problems (Can you determine the interior from boundary measurements?)
• Electrical impedance tomography
• Seismic imaging
• Water wave scattering

In the cluster, the edit Dirichlet-to-Neumann map gives an edit boundary relation.

Applications

• Electrostatics: Finding potential with fixed conductor voltages
• Heat conduction: Steady-state temperature with fixed boundary temperatures
• Gravitational fields: Potential with fixed mass distribution on boundary
• Fluid dynamics: Steady potential flow around objects
• Mathematical physics: Connection to the Riemann mapping theorem, conformal mapping

In the cluster, edit applications include:

• edit Electrostatics
• edit Heat conduction
• edit Gravitational fields
• edit Fluid dynamics
• edit Mathematical physics

This problem

This page is about the Dirichlet problem. nabla^2 f = 0 in V, f = g on S. Uniqueness theorem holds. Solution via Green's function. Method of images for simple geometries. The problem is real.