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.
Hint: Verify the mean value property for your new function obtained by Schwarz reflection principle.
The source adds “cf. Theorem 5.6 on p.60 of Stein et al.” to part (3), a pointer to the Schwarz reflection principle for holomorphic functions in Stein and Shakarchi, Complex Analysis; part (3) asks for the harmonic analogue and does not depend on the reference.