Green's Function of Potential Equation

Goal of using Green’s function is to turn Dirichlet problem of a certain Poisson equation into a special Dirichlet problem, which is only related with the region it concerns and the dimension of the space. By Green’s function, we can represent the solution of any Poisson equation with Dirichlet boundary condition formally.

We first introduce the basic definitions and formulas needed, then construct Green’s function and calculate Green’s function of some usual region, finally discuss weak solutions of PDE briefly.

Preliminaries

Before the main content of this article, we have to review some basic formula we will need, including integration by parts formula, Gauss’s formula and Green’s formulas. We assume is a bounded open and connected subset of satisfying in this section.

Integration by parts formula
Let , then

where is the outer normal vector field of and . Specially, we have

Gauss’s Formula
Let , then we have

and

Green’s Formulas
Let , the Green’s formulas are

and

and

Poisson Equation

Assume region is connected and open, the Poisson equation is defined as

where and are both functions on . If , then the equation deduced into Laplace equation. Poisson equation describes the equilibrium state of the diffusion process. The solutions of Laplace equation are also called harmonic functions. Before considering Poisson equation, we should first investigate the properties of Laplace equation and harmonic functions.

Notice that Laplace operator is invariable under rotation, so it seems that Laplace equation has a radial solution where , is fixed. In fact, the Laplace operator becomes and the solution is

where the constant is chosen to be , is the volume of unit ball in and is the area of unit sphere in . This radial solution is called the fundamental solution of Laplace equation, denoted as .

By fundamental solution of Laplace equation, the solution of Poisson equation turns out to be

since formally we have

in fact, the fundamental solution represents the potential field of a single charge at , and if the charge distribution turns to , all we need to do is just add these fields together and the solution is thus .

Properties of harmonic functions

The most important property of harmonic function is its fundamental integration formula:

which comes from the Green representation of

The second term of Green representation is also called Newton potential with density of .

By using fundamental integration formula we have the powerful averaging formula:

which asserts that value of a harmonic at a point is its average on any ball centering at the same point.

Furthermore, the harmonic functions satisfy maximum principle including strong maximum principle, saying if the maximum point of harmonic function is a inner point then the function can only be constant function, and weak maximum principle, saying the maximum of harmonic function on the whole region is equal with maximum on the boundary . The former is stronger proposition because the latter doesn’t exclude the possibility of maximum point can be inner point.

Green’s Function

In fundamental integration formula of harmonic functions, value of on is unknown, which makes this formula cannot be used to solve Dirichlet problem of Laplace equation directly. However, we can replace this unknown term with by introducing Green’s function.

Using Green’s second formula we have

If we assume satisfy

then can be represented by and . The left of formula above becomes if is also harmonic, and we have

We define is the Green’s function of , then the solution of Dirichlet problem of Laplace equation

is

Similarly, using Green representation of and Green’s second formula we conclude that the solution of Dirichlet problem of Poisson equation

is

Such a formula with Green’s function can be used to analysis the property of the equation as well as its solution, though solving the problem about could be as difficult as the original problem. Fortunately, Green’s functions of some regular regions are easy to calculate and once the Green’s function is obtained, the Dirichlet problem with any boundary condition on the same region can be solved by this method. Besides, the Dirichlet problem of semi-linear equation

can also be investigated by consider the equivalent integration equation:

The restriction of leads to Green’s function have to satisfy

Properties of Green’s function

1. If and , we have ;
2. If , then ;
3. The increasing speed of is when ;
4. Symmetry: ;
5. Normality:.
   And since are harmonic away from , it also has all properties of harmonic functions.

Green’s function of Balls

The Green’s function of ball is

Hence the Dirichlet problem

can be solved by Green’s function:

which is the Poisson integration formula of balls.

Green’s function of Half Upper Space

Denote the half upper space as and the point in is where and , the Green’s function of is

where and . And the Dirichlet problem

can be solved by Green’s function:

which is the Poisson integration formula of half upper space.