- Find a harmonic function \(u\) in the open first quadrant that extends continuously up to the boundary except at the points 0 and 1, and that takes on the following boundary values: \(u(x, y) = 1\) on the half-lines \(\{y = 0, x > 1\}\) and \(\{x = 0, y > 0\}\), and \(u(x, y) = 0\) on the segment \(\{0 < x < 1, y = 0\}\).
[Hint: Find conformal maps \(F_1, F_2, \ldots, F_5\) indicated in Figure 11. Note that \(\frac{1}{\pi}\arg(z)\) is harmonic on the upper half-plane, equals 0 on the positive real axis, and 1 on the negative real axis.]
Solution
Goal: Construct a harmonic function \(u(x, y)\) on the first quadrant \(Q = \{z \in \mathbb{C} : \Re z > 0, \Im z > 0\}\) with the specified boundary values.
1. Conformal mapping of the quadrant to the upper half-plane:
1.1. Let \(Q = \{z \in \mathbb{C} : \Re z > 0, \Im z > 0\}\). Every \(z \in Q\) has polar representation \(z = r e^{i\theta}\) with \(r > 0\) and \(\theta \in (0, \pi/2)\).
1.2. Define \(F(z) = z^2\). For \(z \in Q\), \(F(z) = r^2 e^{i 2\theta}\) with \(2\theta \in (0, \pi)\), so \(F\) maps \(Q\) biholomorphically onto the upper half-plane \(\mathbb{H} = \{w \in \mathbb{C} : \Im w > 0\}\).
1.3. The boundary components of \(Q\) map under \(z \mapsto z^2\) as follows:- The positive imaginary axis \(\{x = 0, y > 0\}\) maps to the negative real axis \((-\infty, 0)\) via \((iy)^2 = -y^2 < 0\).- The segment \(\{0 < x < 1, y = 0\}\) maps to the segment \((0, 1)\) on the real axis via \(x^2 \in (0, 1)\).- The ray \(\{x > 1, y = 0\}\) maps to the ray \((1, \infty)\) on the real axis via \(x^2 \in (1, \infty)\).
2. Construction of the harmonic profile on the upper half-plane \(\mathbb{H}\):
2.1. For \(w \in \mathbb{H}\), let \(\arg(w) = \operatorname{Im}(\log w) \in (0, \pi)\) and \(\arg(w - 1) = \operatorname{Im}(\log(w - 1)) \in (0, \pi)\), where \(\log\) denotes the principal branch of logarithm holomorphic on \(\mathbb{C} \setminus (-\infty, 0]\).
2.2. Both \(\arg(w)\) and \(\arg(w - 1)\) are real parts of holomorphic functions (namely \(\operatorname{Re}(-i \log w)\)), so they are harmonic on \(\mathbb{H}\).
2.3. Define \(v: \mathbb{H} \to \mathbb{R}\) by\[v(w) = 1 - \frac{1}{\pi}\Big(\arg(w - 1) - \arg(w)\Big).\]As a linear combination of harmonic functions, \(v\) is harmonic on \(\mathbb{H}\).
2.4. Boundary limits of \(v(u + iv)\) as \(v \to 0^+\):- For \(u > 1\): \(u > 0\) and \(u - 1 > 0\), so \(\arg(w) \to 0\) and \(\arg(w - 1) \to 0\). Thus \(\lim_{v \to 0^+} v(u + iv) = 1 - \frac{1}{\pi}(0 - 0) = 1\).- For \(0 < u < 1\): \(u > 0\) and \(u - 1 < 0\), so \(\arg(w) \to 0\) and \(\arg(w - 1) \to \pi\). Thus \(\lim_{v \to 0^+} v(u + iv) = 1 - \frac{1}{\pi}(\pi - 0) = 0\).- For \(u < 0\): \(u < 0\) and \(u - 1 < 0\), so \(\arg(w) \to \pi\) and \(\arg(w - 1) \to \pi\). Thus \(\lim_{v \to 0^+} v(u + iv) = 1 - \frac{1}{\pi}(\pi - \pi) = 1\).
3. Pullback to the quadrant \(Q\):
3.1. Define \(u: Q \to \mathbb{R}\) by \(u(z) = v(F(z)) = v(z^2)\).
3.2. Since \(F(z) = z^2\) is holomorphic on \(Q\) and \(v\) is harmonic on \(\mathbb{H}\), the composition \(u = v \circ F\) is harmonic on \(Q\).
3.3. By the boundary correspondence established in 1 and the boundary values of \(v\) from 2:- For \(z = x \in (1, \infty)\), \(z^2 = x^2 \in (1, \infty)\), so \(u(x, 0) = 1\).- For \(z = x \in (0, 1)\), \(z^2 = x^2 \in (0, 1)\), so \(u(x, 0) = 0\).- For \(z = iy\) with \(y > 0\), \(z^2 = -y^2 \in (-\infty, 0)\), so \(u(0, y) = 1\).
4. Conclusion:
The function \[u(z) = 1 - \frac{1}{\pi}\Big(\arg(z^2 - 1) - \arg(z^2)\Big)\] is harmonic in the first quadrant \(Q\), extends continuously to \(\partial Q \setminus \{0, 1\}\), and takes the required boundary values \(1, 0, 1\).