Section 31

Application of Conformal Mappings to Solving Boundary Value Problems for Harmonic Functions

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Problem 31.1

Solve the classical Dirichlet problem for the unit disk {z:∣z∣<1}\left\{ z:\left|z\right| < 1\right\}:

{Δu=0,∣z∣<1;u∣∣z∣=1=12+sin⁡φ, where z=reiφ \begin{cases} \Delta u = 0, & \left|z\right| < 1 ; \\ \left.u\right|_{\left|z\right| = 1} = \frac{1}{2+\sin \varphi }, & \text{ where } z = r e^{i \varphi } \end{cases}
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Problem 31.2

Solve the general Dirichlet problem for the upper half-plane {z:Im(z)>0}\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}:

{Δu=0,y>0u∣y=0=sin⁡x, where z=x+iy \begin{cases} \Delta u = 0, & y > 0 \\ \left.u\right|_{y = 0} = \sin x, & \text{ where } z = x+i y \end{cases}
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Problem 31.3

Solve the general Dirichlet problem for the lower half-plane {z:Im(z)<0}\left\{ z: \mathrm{Im}\left(z\right) < 0\right\}:

{Δu=0,y<0u∣Re(z)⩾0=1,u∣Re(z)<0=0 \begin{cases} \Delta u = 0, & y < 0 \\ \left.u\right|_{\mathrm{Re}\left(z\right) \geqslant 0} = 1, & \left.u\right|_{\mathrm{Re}\left(z\right) < 0} = 0 \end{cases}
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Problem 31.4

Solve the general Dirichlet problem for the exterior of the unit disk {z:∣z∣>1}\left\{ z:\left|z\right| > 1\right\}:

{Δu=0,∣z∣>1u∣∣z∣=1=15−4cos⁡φ \begin{cases} \Delta u = 0, \quad \left|z\right| > 1 \\ \left.u\right|_{\left|z\right| = 1} = \frac{1}{5-4 \cos \varphi } \end{cases}
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Problem 31.5

Solve the general Dirichlet problem for the first quadrant of the plane D={z:Im(z)>0,Re(z)>0}:D = \left\{ z: \mathrm{Im}\left(z\right) > 0, \mathrm{Re}\left(z\right) > 0\right\} :

{Δu=0,z∈D;u∣Re(z)=0Im(z)>0=0,u∣Im(z)=0Re(z)⩾0=1. \begin{cases} \Delta u = 0, & z \in D ; \\ \left.u\right|_{\substack {\mathrm{Re}\left(z\right) = 0 \\ \mathrm{Im}\left(z\right) > 0}} = 0, & \left.u\right|_{\substack {\mathrm{Im}\left(z\right) = 0 \\ \mathrm{Re}\left(z\right) \geqslant 0}} = 1. \end{cases}
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Problem 31.6

Solve the general Dirichlet problem for the half-strip

D={z:0<Im(z)<π,Re(z)>0}:{Δu=0,z∈D;u∣Im(z)=0Re(z)>0=0,u∣Im(z)=πRe(z)>0=0,u∣Re(z)=00⩽Im(z)⩽π=1. D = \left\{ z: 0 < \mathrm{Im}\left(z\right) < \pi , \mathrm{Re}\left(z\right) > 0\right\} : \begin{cases} \Delta u = 0, & z \in D ; \\ \left.u\right|_{\substack {\mathrm{Im}\left(z\right) = 0 \\ \mathrm{Re}\left(z\right) > 0}} = 0, & \left.u\right|_{\substack {\mathrm{Im}\left(z\right) = \pi \\ \mathrm{Re}\left(z\right) > 0}} = 0, & \left.u\right|_{\substack {\mathrm{Re}\left(z\right) = 0 \\ 0 \leqslant \mathrm{Im}\left(z\right) \leqslant \pi }} = 1. \end{cases}
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Problem 31.7

Solve the general Dirichlet problem for the region

D={z:Im(z)>0,∣z+il∣<R}, D = \left\{ z: \mathrm{Im}\left(z\right) > 0,\left|z+i l\right| < R\right\} ,

where l>R>0l > R > 0:

{Δu=0,z∈D;u∣Im(z)=0=0,u∣∣z+il∣=R=T=const. \begin{cases} \Delta u = 0, & z \in D ; \\ \left.u\right|_{\mathrm{Im}\left(z\right) = 0} = 0, & \left.u\right|_{\left|z+i l\right| = R} = T = \mathrm{const}. \end{cases}
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Problem 31.8

Find the Green's function of the Dirichlet problem for the Laplace operator in the unit disk {z:∣z∣<1}\left\{ z:\left|z\right| < 1\right\}.

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Problem 31.9

Find the Green's function of the Dirichlet problem for the Laplace operator in the upper half-plane {z:Im(z)>0}\left\{ z: \mathrm{Im}\left(z\right) > 0\right\}.

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Problem 31.10

Find the Green's function of the Dirichlet problem for the Laplace operator in the unit half-disk {z:∣z∣<1,Im(z)>0}\left\{ z:\left|z\right| < 1, \mathrm{Im}\left(z\right) > 0\right\}.

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