Mathematical Physics Vol 1

7.7 Green formulas

379

The function G , defined in this way, is called the Green function with respect to point A ( a , b , c ) of the sphere S . Observe now a known function H ( x , y , z , a , b , c ) , a fixed point A ( a , b , c ) in region V and an arbitrary point M ( x , y , z ) in this region ( H is a function of A and M ). Assume that this function has the following properties:

1 ◦ it is harmonic with respect to point M ( x , y , z ) ; 2 ◦ it is harmonic with respect to point A ( a , b , c ) ; 3 ◦ its value on the surface S is 1 / r , where r = AM . Let u be a solution of the Dirichlet problem, then, according to (7.229) x S u ∂ H ∂ n − H ∂ u ∂ n d S = 0 ,

(7.283)

and according to 3 ◦ we obtain: x S f

∂ u ∂ n

∂ H ∂ n −

1 r

d S = 0 ⇒

1 4 π x S

1 4 π x S

∂ H ∂ n

∂ u ∂ n

1 r

f

d S =

d S .

(7.284)

Further, according to (7.254), for a "solution" of the Dirichlet problem we obtain

S S

∂ ∂ n ∂ ∂ n

1 r − 1 r −

∂ u ∂ n ∂ u ∂ n

1 4 π x 1 4 π x 1 4 π x S

1 r 1 r

u ( a , b , c )=

f

d S ⇒

u ( a , b , c )=

f

d S ⇒

∂ ∂ n

H d S

1 r −

u ( a , b , c )=

f

(7.285)

Thus, this relation (7.285) gives a solution of the Dirichlet problem, if the function H is known. The function defined by G = H − 1 r (7.286) is called the Green function for region V , with respect to point A ( a , b , c ) . The region is bounded by the surface S . Based on the definition of this function and previous assumptions, we conclude that: 1 ◦ G is a harmonic function in region V , with respect to point M ( x , y , z ) , except at point A ( a , b , c ) ; 2 ◦ G is a harmonic function in region V , with respect to point A ( a , b , c ) , except at point M ( x , y , z ) ; 3 ◦ function G − 1 / r is harmonic at all points of region V ; 4 ◦ on the surface S the function G is annulled. Neumann problem in a plane Let P be a given region in a plane, bounded by the curve ℓ , while on ℓ a function f ( s ) , s ∈ ℓ is defined. The Neumann problem consist of the following: find a function u , which is harmonic in region P , and whose derivative in the direction of the normal ∂ u / ∂ n is a known function f on the contour ℓ , i.e.

∂ u ∂ n

= f ( s ) , s ∈ ℓ.

(7.287)