Integrable Real Functions on Non-Differentiable Curves

In this section we will extend the existence theorem of the Dirichlet problem on the unit disk, given by Theorem 1 in Section 2, to regions bounded by simple closed curves $C_{b}$ which are not differentiable at a finite set of points $z_{b,i}$, for $i\in\{1,\ldots,N\}$. We will still use the conformal transformation known to exist between the open unit disk and the interior of any such curve, as well as its continuous extension to the respective boundaries, where the extension is also differentiable almost everywhere, with the exception of a finite set of singularities of the inverse conformal transformation, at the points $z_{b,i}$, where the inverse conformal transformation still exists but is not differentiable. Note that in Section 4 we established the existence of the conformal mapping $\gamma(z)$ for all simple closed curves $C_{b}$ with finite total lengths, regardless of whether or not the curves are differentiable.

The additional difficulty that appears in this case stems from the fact that, if the curve $C_{b}$ is not differentiable at the points $z_{b,i}$, then the derivative of the transformation $\gamma(z_{a})$ has isolated zeros at the corresponding points $z_{a,i}$ on the curve $C_{a}$, and therefore the derivative of the inverse transformation $\gamma^{(-1)}(z_{b})$ has isolated hard singularities at the points $z_{b,i}$, as was discussed in Section 4. This will require that we impose one additional limitation on the real functions giving the boundary conditions, namely that any integrable hard singularities where they diverge to infinity do not coincide with any of the points $z_{b,i}$.

The precise definition of the Dirichlet problem in this case is the same one given in Definition 3, in Section 5. We will start by establishing the following preliminary fact about the real function $f_{a}(\theta)$ defined from an integrable real function $f_{b}(\lambda)$.

Lemma 2   : Given a real function $f_{b}(\lambda)$ which is integrable on the simple closed curve $C_{b}$ of finite total length, which is not differentiable at a finite set of points $z_{b,i}$, for $i\in\{1,\ldots,N\}$, and given that the integrable hard singularities of $f(\lambda)$ where it diverges to infinity are not located at any of the points $z_{b,i}$ where the curve is non-differentiable, 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 decide whether or not $f_{a}(\theta)$ is also integrable, that is, whether it is integrable on $C_{a}$. Once again, given that $f_{b}(\lambda)$ is integrable on $C_{b}$ we have that the integral

$$\oint_{C_{b}}\vert d\lambda\vert\, \vert f_{b}(\lambda)\vert$$ (32)

exists and is finite. We must now determine whether or not the integral

$$\oint_{C_{a}}\vert d\theta\vert\, \vert f_{a}(\theta)\vert$$ (33)

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 $\lambda$ we obtain once again

$$\oint_{C_{a}}\vert d\theta\vert\, \vert f_{a}(\theta)\vert... ...ac{d\theta}{d\lambda} \right\vert \vert f_{b}(\lambda)\vert.$$ (34)

Since both the absolute value of the derivative shown and the function $f_{b}(\lambda)$ are integrable on $C_{b}$, and since the integrable borderline hard singular points where either one of these two real functions diverges to infinity do not coincide, we have that the integrand of the integral on the right-hand side of this equation is also an integrable real function, and thus that the integral exists and is finite. We can see that the integrand is an integrable real function because around each integrable hard singular point of either one of the two real functions involved there is a neighborhood where the other real function is limited. Since the product of a limited real function with an integrable real function is also an integrable real function, we may conclude that the integrand is locally integrable everywhere on $C_{b}$, and therefore globally integrable there, so that the integral above exists and is finite. It thus follows that $f_{a}(\theta)$ is an integrable real function on $C_{a}$. This establishes Lemma 2.

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

Theorem 4   : Given a simple closed curve $C$ of finite total length on the complex plane $z=x+\mbox{\boldmath$\imath$}y$, which is non-differentiable at a given finite set of points $z_{i}$, for $i\in\{1,\ldots,N\}$, given the conformal transformation $\gamma(z)$ that maps it from the unit circle, whose derivative has zeros on the unit circle at the corresponding set of points, 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 4.1   :

Just as in the previous case, in Section 5, the proof consists of using the conformal transformation between the closed unit disk and the union of the curve $C$ with its interior, 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, thus obtaining the solution of the original Dirichlet problem. 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.

3. The integrable hard singularities of $f(\lambda)$ where it diverges to infinity are not located at any of the points where the curve $C$ is non-differentiable.

The rest of the proof is identical to that of the previous case, in Section 5. Therefore, once again we may conclude that, 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 satisfies $u_{a}(1,\theta)=f_{a}(\theta)$ almost everywhere at the boundary $C_{a}$. Just as in Section 5, we get on the $z_{b}$ plane the complex function $w_{b}(z_{b})$ which 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

$$\nabla^{2}u_{b}(x,y) = 0$$ (35)

in the interior of $C_{b}$, while we also have that

$$u_{b}(x,y)=f_{b}(\lambda),$$ (36)

almost everywhere on $C_{b}$. 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 4.

In this way we have generalized the proof of existence of the Dirichlet problem from the unit circle to all simple closed curves with finite total lengths on the plane, that can be either differentiable or non-differentiable on at most a finite set of points, still for boundary conditions given by integrable real functions.