Mathematical Physics Vol 1

Chapter 7. Partial differential equations

372

Proof By assumption, u and v are harmonic functions, i.e. ∆ u = 0, ∆ v = 0, and thus from (7.229) the statement (7.233) follows directly.

Theorem26 If S is a closed surface, which limits a part of the space V , and U a harmonic function, then y V " ∂ U ∂ x 2 + ∂ U ∂ y 2 + ∂ U ∂ z 2 # dV = x S U · ∂ U ∂ n dS . (7.234)

Proof Let us now apply the second Green formula (7.229), assuming u = U 2 and v ≡ 1. We will first calculate the Laplacian

= 2

∂ U ∂ x

2

∂ 2 u ∂ x 2

∂ 2 U ∂ x 2

∂ u ∂ x

∂ U ∂ x ⇒

= 2 U

+ 2 · U

(7.235)

.

Similarly, we obtain also

= 2

∂ U ∂ y

= 2

∂ U ∂ z

2

2

∂ 2 u ∂ y 2

∂ 2 U ∂ y 2

∂ 2 u ∂ z 2

∂ 2 U ∂ z 2

+ 2 · U

+ 2 · U

(7.236)

,

,

and thus

∆ u = 2 U · ∆ U + 2 "

2 #

∂ U ∂ x

+

∂ U ∂ y

+

∂ U ∂ z

2

2

(7.237)

.

If we now use the assumption that U is a harmonic function, i.e. ∆ U = 0, then from (7.237), for ∆ u , we obtain ∆ u = 2 " ∂ U ∂ x 2 + ∂ U ∂ y 2 + ∂ U ∂ z 2 # . (7.238) Finally, by including (7.238) and v = 1 into (7.229), we obtain y V " ∂ U ∂ x 2 + ∂ U ∂ y 2 + ∂ U ∂ z 2 # d V = x S U ∂ U ∂ n d S , (7.239) which was to be proved. When solving partial differential equations, it can also be required that the solution satisfies specific, the so-called contour conditions 11 , when the values of a function on the surface S (contour - boundary) are known, and we are looking for them in the interior of the region V . Depending on these conditions, we distinguish the following problems: 11 In literature, these conditions are also often called "boundary conditions".