# The $$R^2$$ Dirichlet Problem

Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

## 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

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