Assume \(u\) is harmonic on open set \(O\) and \(z_n\) is a sequence in \(O\) such that \(u(z_n) = 0\) and \(\lim z_n \in O\). Prove or disprove that \(u\) is identically zero. What if \(O\) is a region?
Assume \(u\) is harmonic on open set \(O\) and \(u(z) = 0\) on a disc in \(O\). Prove or disprove that \(u\) is identically zero. What if \(O\) is a region?
Formulate and prove a Schwarz reflection principle for harmonic functions
cf. Theorem 5.6 on p.60 of Stein et al.
Hint: Verify the mean value property for your new function obtained by Schwarz reflection principle.
Solution
Part 1: This is not true: take the holomorphic function \(f(z) = z\), then \(u(z) \da \Re(f(z)) = \Re(z)\) is harmonic on nonzero on \(\RR\) but zero on \(i\RR\).
Part 2: Set \(f \da u_x + i u_y\), then \(f\) is holomorphic on \(O\). Since \(h\equiv 0\) on \(\DD_\eps \subseteq O\), \(g\equiv 0\) on this disc. By the identity principle for holomorphic functions, \(g\equiv 0\) on \(O\). So \(h_x, h_y \equiv 0\), making \(h\) constant, and since \(h\equiv 0\) on \(U\) this forces \(h\equiv 0\) on \(O\).
Part 3: Let \(u\) be harmonic on \(S^+\), a region symmetric about \(\RR\), and that \(u\equiv 0\) on \(\RR \intersect S^+\). Define \(S^- = \ts{\bar{z} \st z\in S^+}\), and \[ U(z) \da \begin{cases} U(z) & z\in S^+ \\ -U(\bar z) & z\in S^-. \end{cases} .\] Then \(U\) is a harmonic extension of \(u\) to \(S \da S^+ \union (S^+ \intersect \RR) \union S^-\). To see that \(U\) is harmonic on \(S\), it suffices to check that \(U\) satisfies the mean value property on \(S\). This holds in \(S^+\): on \(S^+\) we have \(U = u\), and \(u\) is harmonic, so \(u\) satisfies the mean value property there. Thus for \(z_0\in S^+\) we have \[ U(z_0) &= u(z_0) \\ &= {1\over 2\pi} \int_{-\pi}^\pi u(z_0 + re^{it} )\dt \\ &= {1\over 2\pi} \int_{-\pi}^\pi U(z_0 + re^{it} )\dt \\ .\] So for \(w_0\in S^-\), write it as \(w_0 = \bar{z_0}\), then \[ U(z_0) &\da -u(\bar{z_0}) \\ &= {1\over 2\pi }\int_{-\pi}^\pi - u\qty{\bar{z_0 + re^{it} }} \dt \\ &= {1\over 2\pi }\int_{-\pi}^\pi - u\qty{\bar{z_0} + re^{-it} } \dt \\ &= {1\over 2\pi }\int_{-\pi}^\pi - u\qty{\bar{z_0 + re^{-it} }} \dt \\ &= {1\over 2\pi }\int_{-\pi}^\pi U(z_0 + re^{it} ) \dt .\]