Mathematical Physics Vol 1

7.7 Green formulas

373

First contour problem or Dirichlet 12 problem . If the value of the function u ( x , y , z ) on the border S is known, i.e. u ( x , y , z )= f 1 on S , determine its value inside the region V . The function f 1 is known. Second contour problem or Neumann 13 problem . If the derivative of the function u ( x , y , z ) on the surface S is known, i.e. ∂ u ∂ n = f 2 on S , This problem is a combination of the previous two. Namely if the values of a function u ( x , y , z ) and its derivative on the contour S are known, find its value within the region V , i.e. ∂ u ∂ n S + h ( u − f 3 )= 0 . (7.240) Dirichlet problem determine the value of the function u . The function f 2 is known. Third contour problem or mixed contour problem .

Find the solution of the Laplace equation ∆ u = 0 that satisfies the predefined condition

u ( x , y , z )= f ( x , y , z ) on the contour S ,

(7.241)

or shortly

u | S = f . (7.242) In the case of two variables, say x , y , S is a simple closed curve, without singularities.

R Note that this problem can have only one solution.

In order to solve this problem, observe an arbitrary point A ( a , b , c ) of the region V , which belongs to a sub-region Σ ∑ = x , y , z | ( x − a ) 2 +( y − b ) 2 +( z − c ) 2 ≤ R 2 . (7.243) R is determined in such a way that Σ ⊂ V . Let us denote the border of the region Σ as σ .

Figure 7.3: Contour conditions. 12 Peter Gustav Lejeune-Dirichlet (1805-1859), German mathematician. Author of significant papers in analysis, number theory and algebraic structures. He proved the convergence of Fourier series and formulated the general conditions under which a function can be expressed in the form of a trigonometric series. 13 Carl Gottfried Neumann (1832-1925), German mathematician. Known for defining this problem.