Non-Integrable Functions on Non-Differentiable Curves

In this section we will show how one can generalize the proof of existence of the Dirichlet problem on the unit disk, given by Theorem 2 in Section 3, to the case in which we have, as the boundary condition, non-integrable real functions $f_{b}(\lambda)$ defined on boundaries given by non-differentiable curves $C_{b}$ on the plane. We will be able to do this if the non-integrable real functions, despite being non-integrable over the whole curves $C_{b}$, are locally integrable almost everywhere on those curves, and if, in addition to this, the non-integrable hard singularities involved have finite degrees of hardness. The definition of the concept of local integrability almost everywhere is that given by Definition 4, in Section 7. The additional difficulty that appears in this case is the same which was discussed in Section 6, due to the fact that the curve $C_{b}$ is not differentiable at the finite set of points $z_{b,i}$, for $i\in\{1,\ldots,N\}$. Just as in that case, 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 proof will follow the general lines of the one given for integrable real functions in Section 6, with the difference that, just as we did in Section 7, instead of showing that the corresponding functions $f_{a}(\theta)$ on the unit circle $C_{a}$ are integrable there, we will show that they are locally integrable almost everywhere there. In addition to this, instead of using the existence theorem for integrable real function on the unit circle, which was given by Theorem 1 in Section 2, we will use the corresponding result for non-integrable real functions which are locally integrable almost everywhere on the unit circle, which was given by Theorem 2 in Section 3. Since that result depends also on the non-integrable hard singularities of the real functions having finite degrees of hardness, we will also show that the hypothesis that the functions $f_{b}(\lambda)$ have this property implies that the corresponding functions $f_{a}(\theta)$ have the same property as well. In order to do this we will use once again the technique of piecewise integration which was introduced in [#!CAoRFIV!#].

The preliminary result given by Lemma 3 in Section 7 is still valid here. As a consequence of this we may conclude at once that, under the conditions that we have here, the hypothesis that $f_{b}(\lambda)$ is locally integrable almost everywhere on $C_{b}$, with the exclusion of the finite set of points $z_{b,j}$, for $j\in\{1,\ldots,M\}$, implies that $f_{a}(\theta)$ is locally integrable almost everywhere on the unit circle $C_{a}$, with the exclusion of the corresponding finite set of points $z_{a,j}=\gamma^{(-1)}(z_{b,j})$.

We must now discuss the issue of the degrees of hardness of the non-integrable hard singularities of the function $f_{a}(\theta)$ on $C_{a}$. Since by hypothesis $f_{b}(\lambda)$ has non-integrable hard singularities at the points $z_{b,j}$, it clearly follows that $f_{a}(\theta)$ also has hard singularities at the corresponding points $z_{a,j}$, which may be non-integrable ones. In order to discuss their degrees of hardness we will use the technique of piecewise integration, that is, we will consider sectional integrals of $f_{a}(\theta)$ on closed intervals contained within a neighborhood of the point $z_{a,j}$ where it has a single isolated hard singularity. Let us show the following preliminary fact about a real function $f_{a}(\theta)$ defined from a real function $f_{b}(\lambda)$ which is locally integrable almost everywhere on $C_{b}$, and which has non-integrable hard singularities at the finite set of points $z_{b,j}$.

Lemma 5   : Given a real function $f_{b}(\lambda)$ which has an isolated non-integrable hard singularity with finite degree of hardness at a point $z_{b,j}$ on $C_{b}$, and which is such that the hard singularities where it diverges to infinity do not coincide with any of the points $z_{b,i}$ where $C_{b}$ is non-differentiable, it follows that the corresponding function $f_{a}(\theta)$ defined on the unit circle $C_{a}$ by $f_{a}(\theta)=f_{b}(\lambda)$ has an isolated non-integrable hard singularity with finite degree of hardness at the corresponding point $z_{a,j}$ on $C_{a}$.

Since the real functions must diverge to infinity at non-integrable hard singular points, the fact that $f_{a}(\theta)$ has an isolated hard singularity on $z_{a,j}$ is immediate. Since these singularities are all isolated from each other, and since they do not coincide with the points $z_{b,i}$ where $C_{b}$ is non-differentiable, there is on $C_{b}$ a neighborhood of the point $z_{b,j}$ within which $C_{b}$ is differentiable and there are no other non-integrable hard singularities of $f_{b}(\lambda)$. Since the conformal mapping is continuous, it follows that there is on $C_{a}$ a neighborhood of the corresponding point $z_{a,j}$ within which there are no zeros of the derivative of $\gamma(z_{a})$ and no other hard singularities of $f_{a}(\theta)$. The construction of the two sectional primitives of $f_{a}(\theta)$ by means of piecewise integration is the same as the one which was executed before for Lemma 4 in Section 7, resulting in Equation (44). After we change variables from $\theta$ to $\lambda$ on the two sectional integrals in that equation we get

$$f_{a,\ominus}^{-1\prime}(\theta) = \int_{\lambda_{0,\ominus,j}}^{\lambda} d\lambda_{\ominus}\, \frac{d\theta}{d\lambda}(\lambda_{\ominus})\, f_{b}(\lambda_{\ominus}),$$

$$f_{a,\oplus}^{-1\prime}(\theta) = \int_{\lambda_{0,\oplus,j}}^{\lambda} d\lambda_{\oplus}\, \frac{d\theta}{d\lambda}(\lambda_{\oplus})\, f_{b}(\lambda_{\oplus}),$$ (49)

where all the symbols involved are the same as before. Since the derivative $d\theta/d\lambda$ which appears in these integrals is finite, and therefore limited, everywhere within the two lateral intervals involved, due to the fact that the inverse conformal transformation $\gamma^{(-1)}(z_{b})$ is analytic and therefore differentiable within these intervals, it follows that there are two pairs of real numbers $R_{\ominus,m}$ and $R_{\ominus,M}$, as well as $R_{\oplus,m}$ and $R_{\oplus,M}$, such that

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ...ambda}}(\lambda_{\oplus}) & \leq & R_{\oplus,M}, \end{array}$$

everywhere within each interval. Note that, since the conformal transformation, besides being continuous and differentiable, is also invertible within each interval, the derivatives above cannot change sign and thus cannot be zero. Therefore, the pair of bounds $R_{\ominus,m}$ and $R_{\ominus,M}$ may be chosen to have the same sign, and so may the pair of bounds $R_{\oplus,m}$ and $R_{\oplus,M}$. By exchanging the derivative by these extreme values we can obtain upper and lower bounds for the sectional integrals, and therefore we get for the sectional primitives in Equation (53),

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ...da} d\lambda_{\oplus}\, f_{b}(\lambda_{\oplus}). \end{array}$$

We now recognize the integrals that appear in these expressions as the sectional primitives of the function $f_{b}(\lambda)$, so that we get

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ... R_{\oplus,M}\, f_{b,\oplus}^{-1\prime}(\lambda). \end{array}$$

These expressions are true so long as $\varepsilon_{\ominus,j}$ and $\varepsilon_{\oplus,j}$, as well as the corresponding quantities $\delta_{\ominus,j}$ and $\delta_{\oplus,j}$ on $C_{b}$, are not zero, but since the singularities at $z_{b,j}$ are non-integrable hard ones, we cannot take the limit in which these quantities tend to zero. Note that, since $R_{\ominus,m}$ and $R_{\ominus,M}$ have the same sign, and also $R_{\oplus,m}$ and $R_{\oplus,M}$ have the same sign, if the sectional primitives of the function $f_{b}(\lambda)$ diverge to infinity in this limit, then so do the sectional primitives of the function $f_{a}(\theta)$. Therefore, we may conclude that the hard singularity of $f_{a}(\theta)$ is also a non-integrable one. Since we may take further sectional integrals of these expressions, without affecting the inequalities, it is immediately apparent that, after a total of $n$ successive piecewise integrations, we get for the $n^{\rm th}$ sectional primitives

$$<tex2html_comment_mark>\renewedcommand{arraystretch}{2.0} ... ... R_{\oplus,M}\, f_{b,\oplus}^{-n\prime}(\lambda). \end{array}$$

Since the non-integrable hard singularity of $f_{b}(\lambda)$ at the point $\lambda_{j}$ which corresponds to $\theta_{j}$ has a finite degree of hardness, according to the definition of the degrees of hardness, which was given in [#!CAoRFI!#] and discussed in detail for the case of real functions in [#!CAoRFIV!#], there is a value of $n$ such that the limit in which $\delta_{\ominus,j}\to 0$ and $\delta_{\oplus,j}\to 0$ can be taken for the sectional primitives $f_{b,\ominus}^{-n\prime}(\lambda)$ and $f_{b,\oplus}^{-n\prime}(\lambda)$, thus implying that the $n^{\rm th}$ piecewise primitive $f_{b}^{-n\prime}(\lambda)$ of $f_{b}(\lambda)$ is an integrable real function on the whole interval $[\lambda_{\ominus,j},\lambda_{\oplus,j}]$, with a borderline hard singularity, with degree of hardness zero, at the point $\lambda_{j}$. It follows from the inequalities, therefore, that the corresponding limit in which $\varepsilon_{\ominus,j}\to 0$ and $\varepsilon_{\oplus,j}\to 0$ can be taken for the functions $f_{a,\ominus}^{-n\prime}(\theta)$ and $f_{a,\oplus}^{-n\prime}(\theta)$, thus implying that the $n^{\rm th}$ piecewise primitive $f_{a}^{-n\prime}(\theta)$ of $f_{a}(\theta)$ is also an integrable real function on the whole interval $[\theta_{\ominus,j},\theta_{\oplus,j}]$, with a borderline hard singularity, with degree of hardness zero, at $\theta_{j}$. Therefore, the non-integrable hard singularity of $f_{a}(\theta)$ at $\theta_{j}$ has a finite degree of hardness, to wit the same degree of hardness $n$ of the corresponding non-integrable hard singularity of $f_{b}(\lambda)$ at $\lambda_{j}$. This establishes Lemma 5.

We have therefore established that, under the hypothesis that the hard singularities where $f_{b}(\lambda)$ diverges to infinity do not coincide with any of the points $z_{b,i}$ where $C_{b}$ is non-differentiable, so long as $f_{b}(\lambda)$ is locally integrable almost everywhere on $C_{b}$, and so long as its non-integrable hard singularities have finite degrees of hardness, these same two facts are true for $f_{a}(\theta)$ on $C_{a}$. Since we thus see that the necessary properties of the real functions are preserved by the conformal transformation, we are therefore in a position to use the result of Theorem 2 in Section 3 in order to extend the existence theorem of the Dirichlet problem to non-integrable real functions which are, however, integrable almost everywhere on $C_{b}$, this time for the case of non-differentiable curves.

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

Theorem 6   : 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 at the curve.

Proof 6.1   :

Similarly to what was done in the three previous sections, 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 2 in Section 3 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 locally integrable almost everywhere on $C$, including the cases in which this function is globally integrable there.

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 hard singularities of the corresponding inner analytic function $w(z)$ have finite degrees of hardness.

4. The 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 three previous cases. 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 in this case was established in by Theorem 2 Section 3, 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 before, 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

$$\nabla^{2}u_{b}(x,y) = 0$$ (50)

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

$$u_{b}(x,y)=f_{b}(\lambda),$$ (51)

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 6.

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 a finite set of points, but now for boundary conditions given by non-integrable real functions which are locally integrable almost everywhere and have at most a finite set of hard singular points.