Schwarz and Poisson formulas
Fundamental boundary-value problems of potential theory, i.e. (http://planetmath.org/Ie), the so-called Dirichlet and Neumann problems occur in many of applied mathematics such as hydrodynamics, elasticity and electrodynamics. While solving the two-dimensional problem for special of boundaries is likely to present serious computational difficulties, it is possible to write down formulas for a circular (http://planetmath.org/Circle) boundary. We shall give Schwarz and Poisson formulas that solve the Dirichlet problem for a circular domain.
Without loss of generality, we shall consider the compact disc in the plane, its boundary will be denoted by and any point on this one by . Let it be required to determine a harmonic function , which on the boundary assumes the values
where is a continuous single-valued function of . Let be the conjugate harmonic function which is determined to within an arbitrary constant from the knowledge of the function . 11Since is an analytic function of ,it is clear from the Cauchy-Riemann equations that the function is determined by where the integral is evaluated over an arbitrary path joining some point with an arbitrary point belonging to the unitary open disc . We are concerned to a simply connected domain, so that the function will be single-valued.Then the function
is an analytic function for all values of . We shall suppose that the class of continuous functions. Therefore, we may write the boundary condition (1) as
We define here and . Next, we multiply (2) by and, by integrating over , we obtain
which, by Harnack’s theorem, is to (2). Notice that the first integral on the left is equal to by Cauchy’s integral formula, and for the same reason 22From Taylor’s formula But on , , so and term-by-term integration gives the desired result recalling that the second one is equal to . Let , thus (3) becomes
By setting in (4), we get
As one would expect, is left undetermined because the conjugate harmonic function is determined to within an arbitrary real constant. Finally we substitute from (5) in (4),
the aimed Schwarz formula.33It is possible to prove that, if satisfies Hölder condition, then the function given by (6) will be continuous in . Such a condition is less restrictive than the requirement of the existence of a bounded derivative.
If we substitute and in (6) and separate the real and imaginary parts, we find
This is the Poisson formula (so-called also Poisson integral), which gives the solution of Dirichlet problem. It is possible to prove that (7) also the solution under the assumption that is a piecewise continuous function.44See . It is also possible to generalize the formulas obtained above so as to make them apply to any simply connected region. This is done by introducing a mapping function and the idea of conformal mapping of simply connected domains.55For a discussion of Neumann problem, see .
|Title||Schwarz and Poisson formulas|
|Date of creation||2013-03-22 16:05:58|
|Last modified on||2013-03-22 16:05:58|
|Last modified by||perucho (2192)|