Lecture 9. Green's function. Examples

In this chapter we consider the Laplace equation in bounded domains , located on the plane or in space. The points and on the plane (or and in the space) belong to and

(and

) - the distance between points and Let us assume that on the boundary of is set to zero Dirichle condition.

The function is called the Green's function of the Dirichle problem in the , if for any fixed point of it, as a function of , satisfies the following conditions:

a continuous everywhere except at the point , and on the boundary of ;
a harmonic except at point ;
in case the plane is a harmonic function at the point ; if space remains harmonic function at the point .

As follows from the definition of the Green function is continuous and harmonic throughout the domain except point at which it has a feature type in the plane or in space. Green's function is sometimes called the source function.

The Green's function ) (if it exists) is uniquely determined by the properties . In addition, in the domain . Consider, for example, a flat area . To prove the uniqueness of the Green's function, we assume the contrary: let , and - two functions, possessing properties for a given domain and the point . Then remains harmonic at any point in the area , including point , since the vicinity of the point can be written

Each bracket on the right side is a function harmonic everywhere in (see property .), And therefore the difference - everywhere in the harmonic function . Also, at the boundary function Consequently, by the maximum principle in .

Further, if - part of the region , positioned outside a small neighborhood of the point , according to the conditions , the function is continuous in harmonic , and on the border takes non-negative values (as at ). Therefore, on the basis of a maximum of with the zero value inside the area function can not accept. That means that everywhere in .

Example1. In the plane, consider a circle of radius centered at the origin. We construct the Green's function in the circle. In the construction of this function, we need the concept of conjugate points. Points and are conjugate with respect to the circle if they lie on the same ray emanating from the center of the circle , and the product of their distances from the center is equal to the square of the radius: (See Figure16.).

Figure16.

Let and Then Since the points and lie on the same ray emanating from the origin, then

Consider the function

Where (See. Figure17). We verify that it is the Green's function for the circle.

By the theorem of cosines and where

Figure17.

Using the equality , we obtain In this way, the value of and expressed in terms of and, ultimately, through We show that the function satisfies the items determination. It is obvious that the function is continuous everywhere in the closed circle except at the point (when ). At the boundary of a distance and, hence,

Hence The function consists of two terms. The first term, the fundamental solution of the Laplace equation and, therefore, harmonic function everywhere except . The function is harmonic throughout the domain , since the point belongs to the region, and the point lies outside the region , and hence, . The harmony of this function is easily verified if we write the Laplace operator in polar coordinates with pole at (sm. A similar formula with a pole at the point ):

Therefore, the function in harmonic everywhere except at the point Po, and the difference ) - harmonic and at the point .

Similarly, construction of Green's function for a sphere of radius . It has the form Where Point conjugate point with respect to a sphere of radius centered at , that is . The coordinates are calculated according to the formulas:

Example2. Green's function can be viewed not only limited, but also for unbounded domains. As an example, we construct the Green's function for the half-plane. To do this, we define a point conjugate with respect to the line: points and are conjugate with respect to a straight line, if they are symmetrical with respect to this line (see. Figure18).

Figure18. Figure19.

The function where

(See Figure19.), satisfies the properties in the half-plane . In fact, on the boundary at distance , so The harmony function everywhere in the region can be verified directly by calculating the partial derivatives: