Harmonic boundary value problems in half disc and half ring
The authors consider a Poisson kernel which is developed on the basis of the reflection principle leading to the solution of the Schwarz problem for the inhomogeneous Cauchy-Riemann equation for the half unit disc \[ \mathbb{D}^{+}:=\{ z : |z| < 1, 0 < \text{Im}\,\, z \}. \] With the help of the harmonic Green and Neumann functions, the Dirichlet and Neumann boundary value problems are solved explicitly for the Poisson equation in the upper half disc. Particular attention is paid to the two corner points of the boundary for the Dirichlet problem. Also, these boundary value problems are treated in similar manners in the upper half circular ring \[ R^{+} :=\{ z : r < |z| < 1, 0 < \text{Im}\,\,z \}. \] The fundamental tool for complex boundary value problems is the Cauchy-Pompeiu representation, which just has to be properly modified. Theorem. Any \(w \in C^{1} (D;\mathbb{C})\cap C(\overline{D};\mathbb{C})\), for a regular complex domain \(D\subset \mathbb{C}\), can be represented as \[ \displaystyle{ w (z)= \frac{1}{2 \pi i} \displaystyle{\int_{\partial D}^{ }} w (\zeta) \frac{d \zeta}{\zeta - z} - \frac{1}{\pi} \displaystyle{\int_{D}^{ }} w_{\overline{\zeta}}(\zeta) \frac{d \xi d \eta}{\zeta - z}} \] or \[ \displaystyle{ w (z)= \frac{1}{2 \pi i} \displaystyle{\int_{\partial D}} w (\zeta) \frac{d \overline{\zeta}}{\overline{\zeta - z}} - \frac{1}{\pi} \int_{D} w_{\zeta} (\zeta) \frac{d\xi d \eta}{\overline{\zeta - z}}}. \] As one knows, the right-hand sides vanish if \(z\) it is not in \(\overline{D}\), the closure of \(D\). With regard to the Schwarz problem, the Cauchy-Pompeiu formula is modified by combining it in a proper way with the Cauchy-Pompeiu formula evaluated at the reflections of the point \(z\) at the boundary \(\partial D, 1/z\) and \(\overline{z}\), and for symmetry also at the reflection \(1/z\) of \(1/\overline{z}\) at the real axis, or, what is the same, the reflection \(1/z\) of \(\overline{z}\) at the unit circle. Theorem. Any \(w \in C^{1} (\mathbb{D}^{+};\mathbb{C})\cap C(\overline{\mathbb{D}^{+}};\mathbb{C})\) can be represented via \[ \displaystyle{ w (z)= \frac{1}{2 \pi i} \displaystyle{\int_{\partial \mathbf{D}^{+}}} w (\zeta) \frac{d \zeta}{\zeta - z} - \frac{1}{\pi} \displaystyle{\int_{\mathbf{D}^{+}}} w_{\overline{\zeta}}(\zeta) \frac{d \xi d \eta}{\zeta - z}}, z\in \mathbb{D}^{+}, \] and for \(z\in \mathbb{D}^{+}\) \[ \displaystyle{ w (z)= \frac{1}{2 \pi i} \displaystyle{\int_{|\zeta|=1,\, {0 < Im \, \zeta}}} \text{Re} \,\, w(\zeta) \left [ \frac{\zeta + z}{\zeta - z} - \frac{\overline{\zeta} + z}{\overline{\zeta - z}} \right ] \frac{d\zeta}{\zeta} + \frac{1}{\pi} \int_{|\zeta|=1, \, 0 < \text{Im }\, \zeta}\text{Im} \,\, w (\zeta) \frac{d \zeta}{\zeta} } \] \[ \displaystyle{+ \frac{1}{\pi i} \int^{1}_{-1} \text{Re} \,\, w(t) \left [ \frac{1}{t-z} - \frac{z}{1-zt} \right ]dt } \] \[ \displaystyle{- \frac{1}{\pi i} \int_{\mathbf{D}^{+}} \left \{ w_{\overline{\zeta}}(\zeta) \left [ \frac{1}{\zeta - z} - \frac{z}{1-z \zeta} \right ] - \overline{w_{\overline{\zeta}} (\zeta)} \left [ \frac {1}{\overline{\zeta} - z}- \frac {z}{1 - z \overline{\zeta}} \right ] \right \} d \xi d \eta }. \] This formula provides the solution to the Schwarz problem for \(\mathbb{D}^{+}.\) Theorem. The Schwarz problem \[ w_{\overline{z}}=f \text{ in } \mathbb{D}^{+}, \,\,\,\, f \in L_{p} (\mathbb{D}^{+}; \mathbb{C}), \,\,\, p > 2, \] \[ \text{Re} \, w = \gamma \text{ in } \partial \mathbb{D}^{+}, \,\,\,\, \gamma \in C (\partial \mathbb{D}^{+}; \mathbb{C}), \,\,\,\, \gamma(-1) = \gamma(1) = 0, \] \[ \displaystyle{\frac{1}{\pi} \int_{0}^{\pi} Im \,\, w(e^{i \varphi}) d\varphi = c, \,\,\,\, c \in \mathbb{R} }, \] is uniquely solvable by \[ \displaystyle{w(z) = \frac{1}{2 \pi i} \int_{|\zeta| = 1, \, 0 < Im \, \zeta} \gamma(\zeta) \left [ \frac{\zeta + z}{\zeta - z} - \frac{\overline{\zeta} + z}{\overline{\zeta} - z} \right ] \frac{d \zeta}{\zeta} + i c + \frac{1}{\pi i} \int_{-1}^{1} \gamma(t) \left [ \frac{1}{t-z} - \frac{z}{1-zt} \right ] dt } \] \[ \displaystyle{- \frac{1}{\pi} \int_{\mathbf{D}^{+}} \left \{f(\zeta) \left [ \frac{1}{\zeta - z} - \frac{z}{1 - z \zeta} \right ] - \overline{f(\zeta)} \left [ \frac{1}{\overline{\zeta} - z} - \frac{z}{1 - z \overline{\zeta}} \right ] \right \} d \xi d \eta }. \] Iterating the Cauchy-Pompeiu formulas leads to second order representations. Introducing the harmonic Green function in the respective formulas with regard to the Laplace operator, gives a representation adjusted to the Dirichlet problem for the Poisson equation. Theorem. Let \(D \subset \mathbb{C}\) be a regular domain, and let \(G_{1} = 2G\) where \(G\) is the harmonic Green function for \(D\). Then any \(w \in C^{2}(D;\mathbb{C}) \cap C^{1}(\overline{D};\mathbb{C})\) can be represented by \[ \displaystyle{w(z) = - \frac{1}{4 \pi} \int_{\partial D} w(\zeta) \partial_{v_{\zeta}} G_{1} (z, \zeta)ds_{\zeta} - \frac{1}{\pi} \int_{D} w_{\zeta \overline{\zeta}} G_{1} (z, \zeta) d \xi d \eta }. \] Here \(v\) is the outward normal derivative on \(\partial D\), and \(s\) is the arc length parameter. Theorem. The Dirichlet problem \[ w_{z \overline{z}} = f \,\,\,\, in\,\, \mathbb{D}^{+}, \,\,\,\, w = \gamma \text{ on } \partial \mathbb{D}^{+}, \] for \(f \in C(\mathbb{D}^{+}; \mathbf{C}), \, \gamma \in C(\partial \mathbb{D}^{+}; \mathbb{C}), \, \gamma(-1) = \gamma(1) = 0\), is uniquely solvable by \[ \displaystyle{w(z) = \frac{1}{2 \pi i} \int_{|\zeta| = 1, \, 0 < Im \, \zeta} \gamma (\zeta) \left [ \frac{\zeta}{\zeta - z} + \frac{\overline{\zeta}} {\overline{\zeta - z}} - \frac{\zeta}{\zeta - \overline{z}} - \frac{\overline{\zeta}}{\overline{\zeta} - z} \right ] \frac{d \zeta}{\zeta}} \] \[ \displaystyle{+ \frac{1}{2 \pi i} \int_{-1}^{1} \gamma (t) \left [ \frac{z - \overline{z}}{|t-z|^{2}} - \frac{z - \overline{z}}{|1 - tz|^{2}} \right ] dt - \frac{1}{\pi} \int_{\mathbb{D}^{+}} f(\zeta) G_{1} (z, \zeta) d \xi d \eta }. \] Another representation formula of second order besides the one in the second above presented Theorem is available, where instead of the Green function the harmonic Neumann function is used. Theorem. Let \(D \subset \mathbb{C}\) be a regular domain and \(N_{1} = 2N\) where \(N\) is the harmonic Neumann function for \(D\). Then any \(w \in C^{2}(D;\mathbb{C}) \cap C^{1}(\overline{D}; \mathbb{C})\) can be represented as \[ \displaystyle {w(z) = - \frac{1}{4 \pi} \int_{\partial D} \{w(\zeta) \partial_{v_{\zeta}} N_{1} (z, \zeta) - \partial_{v_{\zeta}} w(\zeta) N_{1}(z, \zeta)\} ds_{\zeta} - \frac{1}{\pi} \int_{D} w_{\zeta \overline{\zeta}} (\zeta)N_{1} (z, \zeta) d \xi d \eta }. \] Theorem. The Neumann problem \[ w_{z \overline{z}} = f \,\,\,\, in\,\, \mathbb{D}^{+}, \,\,\,\,\,\, \partial_{v} w = \gamma \text{ on } \partial \mathbb{D}^{+}, \] for \(f \in C(\mathbb{D}^{+}; \mathbb{C}), \,\, \gamma \in C(\partial \mathbb{D}^{+}; \mathbb{C}),\) is uniquely solvable. The solution is given by \[ \displaystyle{w(z) = \frac{1}{2 \pi i} \int_{|\zeta| = 1, \, 0 < Im \, \zeta} \gamma(\zeta) [ \text{log\,} |z|^{2} - \text{log}\, |(1 - z \overline{\zeta})(1 - z \zeta)|^{2} ] \frac{d \zeta}{\zeta} } \] \[ \displaystyle{+ \frac{1}{2 \pi} \int_{-1}^{1} \gamma(t) [ \text{log\,} |tz|^{2} -\text{log\,} |(t - z)(1 - zt)|^{2} ] dt } \] \[ \displaystyle{- \frac{1}{\pi} \int_{\mathbf{D}^{+}} f(\zeta) [ 2 \,\, \text{log\,} |z \zeta|^{2} - \text{log\,} |(\zeta - z)(\overline{\zeta} - z)(1 - z \overline{\zeta})(1 - z \zeta)|^{2} ] d \xi d \eta }. \] The authors give also six very interesting similarly results for the case of the upper half circular ring.
scientific article; zbMATH DE number 5620898