Let us consider the interior Dirichlet problem for the Laplace equation in a starlike annulus

$\mathcal{A}$, whose boundaries

${\partial}_{\pm}\mathcal{A}$ are described by the polar equations

$r={R}_{\pm}(\vartheta )$ respectively

Under the mentioned assumptions, one can prove the following theorem.

**Theorem**
*Let*
${f}_{\pm}({R}_{\pm}(\vartheta )cos\vartheta ,{R}_{\pm}(\vartheta )sin\vartheta )={F}_{\pm}(\vartheta )=\sum _{m=0}^{+\mathrm{\infty}}({\alpha}_{m}^{(\pm )}cosm\vartheta +{\beta}_{m}^{(\pm )}sinm\vartheta ),$

(13)

*where*
$\left\{\begin{array}{c}{\alpha}_{m}^{(\pm )}\\ {\beta}_{m}^{(\pm )}\end{array}\right\}=\frac{{\u03f5}_{m}}{2\pi}{\int}_{0}^{2\pi}{F}_{\pm}(\vartheta )\left\{\begin{array}{c}cosm\vartheta \\ sinm\vartheta \end{array}\right\}\phantom{\rule{0.2em}{0ex}}d\vartheta ,$

(14)

${\u03f5}_{m}$ *being the usual Neumann symbol*.

*Then the boundary*-

*value problem* (12)

*for the Laplace equation admits a classical solution* $v(x,y)\in {L}^{2}(\mathcal{A})$ *such that the following Fourier*-

*like series expansion holds true*:

$\begin{array}{c}v(\frac{(b-\varrho ){R}_{-}(\vartheta )-(a-\varrho ){R}_{+}(\vartheta )}{b-a}cos\vartheta ,\frac{(b-\varrho ){R}_{-}(\vartheta )-(a-\varrho ){R}_{+}(\vartheta )}{b-a}sin\vartheta )\hfill \\ \phantom{\rule{1em}{0ex}}=U(\varrho ,\vartheta )\hfill \\ \phantom{\rule{1em}{0ex}}=\sum _{m=-\mathrm{\infty}}^{+\mathrm{\infty}}{\left[\frac{(b-\varrho ){R}_{-}(\vartheta )-(a-\varrho ){R}_{+}(\vartheta )}{b-a}\right]}^{m}({A}_{m}cosm\vartheta +{B}_{m}sinm\vartheta )\hfill \\ \phantom{\rule{2em}{0ex}}+{\delta}_{0}ln\left(\frac{(b-\varrho ){R}_{-}(\vartheta )-(a-\varrho ){R}_{+}(\vartheta )}{b-a}\right).\hfill \end{array}$

(15)

*For each index* *m*,

*define* $\left[\begin{array}{c}{\xi}_{m}^{(\pm )}(\vartheta )\\ {\eta}_{m}^{(\pm )}(\vartheta )\end{array}\right]={R}_{\pm}(\vartheta )\left[\begin{array}{c}cosm\vartheta \\ sinm\vartheta \end{array}\right],$

(16)

*and set*,

*for shortness*,

${\zeta}^{(\pm )}(\vartheta )=ln{R}_{\pm}(\vartheta )$.

*In this way*,

*the coefficients* ${\delta}_{0}$ *and* ${A}_{m}$,

${B}_{m}$ *appearing in* (15)

*can be determined by solving the infinite linear system* $\sum _{m=-\mathrm{\infty}}^{+\mathrm{\infty}}\left[\begin{array}{cc}{\mathrm{X}}_{{\mathrm{C}}_{n,m}}^{(-)}& {\mathrm{Y}}_{{\mathrm{C}}_{n,m}}^{(-)}\\ {\mathrm{X}}_{{\mathrm{S}}_{n,m}}^{(-)}& {\mathrm{Y}}_{{\mathrm{S}}_{n,m}}^{(-)}\\ {\mathrm{X}}_{{\mathrm{C}}_{n,m}}^{(+)}& {\mathrm{Y}}_{{\mathrm{C}}_{n,m}}^{(+)}\\ {\mathrm{X}}_{{\mathrm{S}}_{n,m}}^{(+)}& {\mathrm{Y}}_{{\mathrm{S}}_{n,m}}^{(+)}\end{array}\right]\cdot \left[\begin{array}{c}{A}_{m}\\ {B}_{m}\end{array}\right]+\left[\begin{array}{c}{\mathrm{Z}}_{{\mathrm{C}}_{n}}^{(-)}\\ {\mathrm{Z}}_{{\mathrm{S}}_{n}}^{(-)}\\ {\mathrm{Z}}_{{\mathrm{C}}_{n}}^{(+)}\\ {\mathrm{Z}}_{{\mathrm{S}}_{n}}^{(+)}\end{array}\right]{\delta}_{0}=\left[\begin{array}{c}{\alpha}_{n}^{(-)}\\ {\beta}_{n}^{(-)}\\ {\alpha}_{n}^{(+)}\\ {\beta}_{n}^{(+)}\end{array}\right],$

(17)

*where*
${\mathrm{X}}_{{\left\{\begin{array}{c}\mathrm{C}\\ \mathrm{S}\end{array}\right\}}_{n,m}}^{(\pm )}=\frac{{\u03f5}_{n}}{2\pi}{\int}_{0}^{2\pi}{\xi}_{m}^{(\pm )}(\vartheta )\left\{\begin{array}{c}cosn\vartheta \\ sinn\vartheta \end{array}\right\}\phantom{\rule{0.2em}{0ex}}d\vartheta ,$

(18)

${\mathrm{Y}}_{{\left\{\begin{array}{c}\mathrm{C}\\ \mathrm{S}\end{array}\right\}}_{n,m}}^{(\pm )}=\frac{{\u03f5}_{n}}{2\pi}{\int}_{0}^{2\pi}{\eta}_{m}^{(\pm )}(\vartheta )\left\{\begin{array}{c}cosn\vartheta \\ sinn\vartheta \end{array}\right\}\phantom{\rule{0.2em}{0ex}}d\vartheta ,$

(19)

${\mathrm{Z}}_{{\left\{\begin{array}{c}\mathrm{C}\\ \mathrm{S}\end{array}\right\}}_{n}}^{(\pm )}=\frac{{\u03f5}_{n}}{2\pi}{\int}_{0}^{2\pi}{\zeta}^{(\pm )}(\vartheta )\left\{\begin{array}{c}cosn\vartheta \\ sinn\vartheta \end{array}\right\}\phantom{\rule{0.2em}{0ex}}d\vartheta ,$

(20)

*with* $m\in \mathbb{Z}$, *and* $n\in {\mathbb{N}}_{0}:=\mathbb{N}\cup \{0\}$.

*Proof* Upon noting that in the stretched coordinate system

$\varrho ,\vartheta $ introduced in the

*x*,

*y* plane, the considered domain

$\mathcal{A}$ turns into the circular annulus of radii

*a* and

*b*, one can readily adopt the usual eigenfunction method [

11] in combination with the separation of variables (with respect to

*r* and

*ϑ*). As a consequence, elementary solutions of the problem can be searched in the form

$u(r,\vartheta )=U(\frac{b[r-{R}_{-}(\vartheta )]-a[r-{R}_{+}(\vartheta )]}{{R}_{+}(\vartheta )-{R}_{-}(\vartheta )},\vartheta )=\mathrm{P}(r)\mathrm{\Theta}(\vartheta ).$

(21)

Substituting into the Laplace equation, one easily finds that the functions

$\mathrm{P}(\cdot )$,

$\mathrm{\Theta}(\cdot )$ must satisfy the ordinary differential equations

$\frac{{d}^{2}\mathrm{\Theta}(\vartheta )}{d{\vartheta}^{2}}+{\mu}^{2}\mathrm{\Theta}(\vartheta )=0,$

(22)

${r}^{2}\frac{{d}^{2}\mathrm{P}(r)}{d{r}^{2}}+r\frac{d\mathrm{P}(r)}{dr}-{\mu}^{2}\mathrm{P}(r)=0,$

(23)

respectively. The parameter

*μ* is a separation constant whose choice is governed by the physical requirement that at any fixed point in the real plane the scalar field

$u(r,\vartheta )$ must be single-valued. So, by setting

$\mu =m\in {\mathbb{N}}_{0}$, one can easily find

$\mathrm{\Theta}(\vartheta )={a}_{m}cosm\vartheta +{b}_{m}sinm\vartheta ,$

(24)

where

${a}_{m},{b}_{m}\in \mathbb{C}$ denote arbitrary constants. The radial function

$\mathrm{P}(\cdot )$ satisfying (23) can be readily expressed as follows:

$\mathrm{P}(r)=\{\begin{array}{ll}{c}_{m}{r}^{m}+{d}_{m}{r}^{-m},& m\ne 0,\\ {c}_{0}+{d}_{0}lnr,& m=0,\end{array}$

(25)

with

${c}_{m},{d}_{m}\in \mathbb{C}$. Therefore, the general solution of the Dirichlet problem (12) can be searched in the form

$u(r,\vartheta )=\sum _{m=-\mathrm{\infty}}^{+\mathrm{\infty}}{r}^{m}({A}_{m}cosm\vartheta +{B}_{m}sinm\vartheta )+{\delta}_{0}lnr.$

(26)

Enforcing the Dirichlet boundary condition readily yields ${F}_{\pm}(\vartheta )=u({R}_{\pm}(\vartheta ),\vartheta )$. Hence, using the classical Fourier projection method, equations (17)-(20) follow after some trivial manipulations. □

It is worth noting that the derived expressions still hold under the assumption that ${R}_{\pm}(\vartheta )$ are piecewise continuous functions, and the boundary values are described by square integrable, not necessarily continuous, functions so that the relevant Fourier coefficients ${\alpha}_{m}^{(\pm )}$, ${\beta}_{m}^{(\pm )}$ in equation (14) are finite quantities.