# The \(R^2\) Dirichlet Problem

## Abstract

It is shown how an approximate solution to a Dirichlet problem-a real-valued continuous function g being given on the boundary of a bounded domain \(D\) without holes-can be constructed using any one function \(f(z)\), not a polynomial, which has a convergent power series at some point. (And it is shown that if \(f\) is a polynomial, the result is false.) The proof uses the Hahn-Banach Theorem. If the function \(f\) is taken to have a power series convergent about zero, the solution, harmonic in \(D\) and approximating \(g\) on the boundary, is the real part of a finite sum of terms of the form \({c_n}(f({a_n}(z + {b_n}))\). In applications the real parameters \(a_n\), \({b_n}\), \({c_n}\), \(n\) = 1, 2, . . . ,\(M\) are obtained by computer minimization of the error of the fit to the given function \(g\) on the boundary of the domain. The proof depends on the Walsh-Lebesgue Theorem, which is shown to be equivalent to the complex variable approximate boundary value result as stated in this chapter. Note that the approximation is to the boundary function \(g\) on the boundary of the domain, the approximating functions satisfy the Laplace equation exactly in \(D\) .

### Keywords

Hole in a domain Walsh-Lebesgue Theorem C(\(\Gamma \)) Hahn-Banach Theorem.### References

- 1.Ransford T (1995) Potential theory in the complex Plane. Cambridge University Press, CambridgeGoogle Scholar
- 2.Whitley R, Hromadka T II (2001) A general complex variable boundary element method, Numer Methods Partial Differ Equ 17:332–335Google Scholar
- 3.Hromadka T II, Lai C (1987) The complex variable boundary element method in engineering analysis, Springer, New YorkGoogle Scholar
- 4.Hromadka T II, Whitley R (1998) Advances in the complex variable boundary element method, Springer, New YorkGoogle Scholar