Mathematical Physics Vol 1

7.7 Green formulas

375

Finally, as

∂ ∂ n

1 r

1 r 2

(7.253)

= −

we obtain

S

∂ ∂ n

1 r −

∂ u ∂ n

1 4 π x

1 r

u ( a , b , c )=

f

d S .

(7.254)

This relation, for now, does not give the solution of the problem, as there exists an unknown value ∂ u / ∂ n under the integral.

R Note that we could solve the problem in a similar way when the function depends on two variables, x , y . In this case, we would replace the surface with a line l , and also replace σ = 4 π R 2 by l = 2 π R . Let us solve the previous problem (Dirichlet problem) in two special cases, when the region Σ is: • a circle and • a sphere. Dirichlet problem for a circle The task is to find a function u that satisfies the Laplace equation ∆ u = 0 within the circle K = ( x , y ) | x 2 + y 2 = R 2 0 . Due to the nature of the problem, it is more convenient tot use polar coordinates ( r , ϕ ) , in which the Laplace equation takes the form

∂ 2 u ∂ r 2

∂ 2 u ∂ϕ 2

∂ u ∂ r

+ r 2

r

= 0 .

(7.255)

+

Let us look for a solution in the form

u = R ( r ) · F ( ϕ ) ,

(7.256)

and thus the equation (7.255) becomes

r 2 R ′′ + r · R ′ F + R · F ′′ = 0 ,

(7.257)

that is

r 2 · R ′′ + r · R ′ R

F ′′ F

= k 2 ,

(7.258)

= −

where k is a constant. We shall analyze two cases: k̸ = 0 and k = 0. For k̸ = 0 we obtain:

r 2 R ′′ + r · R ′ − R · k 2 = 0 - Euler equation and F ′′ + k 2 F = 0 - diff. eq. with const. coefficients.

(7.259) (7.260)

The solutions of these equations are

k + C

k , F ( ϕ )= C

3 cos ( k ϕ )+ C 4 sin ( k ϕ ) ,

2 r −

R ( r )= C 1 r

(7.261)

where C i ( i = 1 , 2 , 3 , 4) are arbitrary constants.