Integrable Real Functions on Differentiable Curves

In this section we will show how one can generalize the proof of existence of the solution of the Dirichlet problem on the unit disk, given by Theorem 1 in Section 2, to the case in which we have, as the boundary condition, integrable real functions defined on boundaries given by differentiable curves on the plane. In order to do this, the first thing we must do here is to establish the precise definition of the Dirichlet problem in this case.

Definition 3   : The Dirichlet Problem on a Given Curve and its Interior

Given a simple closed curve $C$ on the complex plane, described by a real arc-length parameter $\lambda$, and a real function $f(\lambda)$ defined on it, the existence problem of the Dirichlet boundary value problem of the Laplace equation on this curve and its interior is to show that a function $u(x,y)$ exists such that it satisfies

$$\nabla^{2}u(x,y) = 0,$$ (20)

within the interior of $C$, and such that it also satisfies

$$u(x,y) = f(\lambda),$$ (21)

for $z=x+\mbox{\boldmath$\imath$}y$ on $C$, thus corresponding to $\lambda$, almost everywhere over that curve.

Therefore, using the notation established in Section 4, let $C_{b}$ be a differentiable simple closed curve on the complex $z_{b}$ plane, with finite total length, and let us assume that a Dirichlet boundary value problem for the Laplace equation is given on the region whose boundary is the curve $C_{b}$, that is, let there be given also an integrable real function $f_{b}(\lambda)$ on $C_{b}$, that is, a function such that the integral

$$\oint_{C_{b}}d\lambda\, f_{b}(\lambda)$$ (22)

exists and is finite. The problem is then to establish the existence of a function $u_{b}(x,y)$ that satisfies $\nabla^{2}u_{b}(x,y)=0$ in the interior of the curve $C_{b}$ and that assumes the values $f_{b}(\lambda)$ almost everywhere over that curve. In order to do this using the conformal transformation $\gamma(z)$ from the complex plane $z_{a}$ to the complex plane $z_{b}$, we start by constructing a corresponding Dirichlet problem on the unit disk of the $z_{a}$ plane, using the mapping between $z_{b}$ and $z_{a}$ provided by the conformal transformation $\gamma(z_{a})$ and its inverse $\gamma^{(-1)}(z_{b})$. We define a corresponding real function $f_{a}(\theta)$ on the unit circle $C_{a}$ by simply transferring the values of $f_{b}(\lambda)$ through the use of the conformal mapping from point to point,

$$f_{a}(\theta) = f_{b}(\lambda)$$

$$= f_{b}(g(\theta)),$$ (23)

where $\theta$ describes a point on $C_{a}$ given by the complex number $z_{a}$, $\lambda$ describes the corresponding point on $C_{b}$ given by the complex number $z_{b}=\gamma(z_{a})$, and $g(\theta)$ is the induced real transformation, so that we have that $\lambda=g(\theta)$ and that $\theta=g^{(-1)}(\lambda)$. We will start by establishing the following preliminary fact about the real function $f_{a}(\theta)$ defined as above from an integrable real function $f_{b}(\lambda)$.

Lemma 1   : Given a real function $f_{b}(\lambda)$ which is integrable on the differentiable simple closed curve $C_{b}$ of finite total length, it follows that the corresponding function $f_{a}(\theta)$ defined on the unit circle $C_{a}$ by $f_{a}(\theta)=f_{b}(\lambda)$ is integrable on that circle.

Given that $f_{b}(\lambda)$ is integrable on $C_{b}$, we must show that $f_{a}(\theta)$ defined by the composition of $f_{b}(\lambda)$ with $g(\theta)$, is also integrable, that is, it is integrable on $C_{a}$. Since all real functions under discussion here are assumed to be Lebesgue-measurable, and since for such measurable functions defined on compact domains integrability and absolute integrability are equivalent conditions [#!RARudin!#,#!RARoyden!#], given that $f_{b}(\lambda)$ is integrable on $C_{b}$ we have that the integral

$$\oint_{C_{b}}\vert d\lambda\vert\, \vert f_{b}(\lambda)\vert$$ (24)

exists and is finite. We must now show that the integral

$$\oint_{C_{a}}\vert d\theta\vert\, \vert f_{a}(\theta)\vert$$ (25)

exists and is finite, which is equivalent to the statement that $f_{a}(\theta)$ is integrable on $C_{a}$. Changing variables on this integral from $\theta$ to $\lambda$, and using the fact that by definition we have that $f_{a}(\theta)=f_{b}(\lambda)$, we obtain

$$\oint_{C_{a}}\vert d\theta\vert\, \vert f_{a}(\theta)\vert... ...ac{d\theta}{d\lambda} \right\vert \vert f_{b}(\lambda)\vert.$$ (26)

Since $f_{b}(\lambda)$ is integrable on $C_{b}$, and since the absolute value of the derivative shown exists and is finite, given that we have

$$\left\vert \frac{d\theta}{d\lambda} \right\vert = \left\vert \frac{dz_{a}}{dz_{b}} \right\vert$$

$$= \left\vert \frac{d\gamma^{(-1)}(z_{b})}{dz_{b}} \right\vert,$$ (27)

where $\gamma^{(-1)}(z_{b})$ is analytic on $C_{b}$ and therefore differentiable there, it follows that the absolute value of the derivative which appears in the integrand on the right-hand side of Equation (26) is a limited real function on $C_{b}$. Since $f_{b}(\lambda)$ is integrable on $C_{b}$, from this it follows that the whole integrand of the integral on the right-hand side of Equation (26), which is the product of a limited real function with an integrable real function, is itself an integrable real function on $C_{b}$, so that we may conclude that the integral in Equation (25) exists and is finite, and therefore that $f_{a}(\theta)$ is an integrable real function on $C_{a}$. This establishes Lemma 1.

In this section, using the results from the previous sections, we will establish the following theorem.

Theorem 3   : Given a differentiable simple closed curve $C$ of finite total length on the complex plane $z=x+\mbox{\boldmath$\imath$}y$, given the invertible conformal transformation $\gamma(z)$ whose derivative has no zeros on the closed unit disk, that maps it from the unit circle, and given a real function $f(\lambda)$ on that curve, that satisfies the list of conditions described below, there is a solution $u(x,y)$ of the Dirichlet problem of the Laplace equation within the interior of that curve, that assumes the given values $f(\lambda)$ almost everywhere on the curve.

Proof 3.1   :

The proof consists of using the conformal transformation between the closed unit disk and the union of the curve $C$ with its interior, a transformation which according to the analysis in Section 4 always exists, to map the given boundary condition on $C$ onto a corresponding boundary condition on the unit circle, then using the proof of existence established before by Theorem 1 in Section 2 for the closed unit disk to establish the existence of the solution of the corresponding Dirichlet problem on that disk, and finally using once more the conformal transformation to map the resulting solution back to $C$ and its interior, showing in the process that one obtains in this way the solution of the Dirichlet problem there. The list of conditions on the real functions is now the following.

1. The real function $f(\lambda)$ is integrable on $C$.

2. The number of hard singularities on the unit circle of the corresponding inner analytic function $w(z)$ on the unit disk is finite.

According to the preliminary result established in Lemma 1, if the real function $f_{b}(\lambda)$ satisfies these conditions on $C_{b}$, then $f_{a}(\theta)$ is an integrable real function on $C_{a}$. Therefore, due to the existence theorem of the Dirichlet problem on the unit disk of the plane $z_{a}$, which was established by Theorem 1 in Section 2, we know that there is an inner analytic function $w_{a}(z_{a})$ such that its real part $u_{a}(\rho,\theta)$ is harmonic within the open unit disk and also satisfies $u_{a}(1,\theta)=f_{a}(\theta)$ almost everywhere at the boundary $C_{a}$. Now, by composing $w_{a}(z_{a})$ with the inverse conformal transformation $\gamma^{(-1)}(z_{b})$ we get on the $z_{b}$ plane the complex function

$$w_{b}(z_{b}) = w_{a}(z_{a})$$

$$= w_{a}\!\left(\gamma^{(-1)}(z_{b})\right),$$ (28)

which corresponds to simply transferring back the values of $w_{a}(z_{a})$, by the use of the conformal mapping from point to point, while we also have, of course, the corresponding inverse real transformation at the boundary,

$$f_{b}(\lambda) = f_{a}(\theta)$$

$$= f_{a}\!\left(g^{(-1)}(\lambda)\right).$$ (29)

Given that the composition of two analytic functions is also analytic, in their chained domain of analyticity, and since $\gamma^{(-1)}(z_{b})$ is analytic in the interior of the curve $C_{b}$, and also since $w_{a}(z_{a})$ is analytic in the interior of the curve $C_{a}$, we conclude that $w_{b}(z_{b})$ is analytic in the interior of the curve $C_{b}$. Therefore, the real part $u_{b}(x,y)$ of $w_{b}(z_{b})$ is harmonic and thus satisfies

$$\nabla^{2}u_{b}(x,y) = 0$$ (30)

in the interior of $C_{b}$, while by construction the fact that we have $u_{a}(\rho,\theta)=f_{a}(\theta)$ for the points given by $z_{a}=\rho\exp(\mbox{\boldmath$\imath$}\theta)$ almost everywhere on $C_{a}$, where $\rho=1$, implies that we also have

$$u_{b}(x,y)=f_{b}(\lambda),$$ (31)

for the corresponding points given by $z_{b}=x+\mbox{\boldmath$\imath$}y$ almost everywhere on $C_{b}$, and which thus correspond to $\lambda$. This establishes the existence, by construction, of the solution of the Dirichlet problem on the $z_{b}$ plane, under our current hypotheses. This completes the proof of Theorem 3.

In this way we have generalized the proof of existence of the Dirichlet problem from the unit circle to all differentiable simple closed curves with finite total lengths on the plane, for boundary conditions given by integrable real functions.