MATH3401
Complex Analysis
Lecture 26
Dirichlet and Neumann problems
\(\,f\) conformal. \(C\) is a smooth (infinitely differentiable) arc in \(\Omega\), and \(\Gamma = f(C)\).
As before, take \(H(x,y) = h\left(u(x,y), v(x,y)\right)\).
Dirichlet and Neumann problems
1) If we have Dirichlet (boundary) conditions on \(\Gamma\), i. e. \(h(u,v) = \varphi\) on \(\Gamma\), then \[H(x,y) = \varphi \text{ on } C.\]
2) If we have homogeneous Neumann boundary conditions on \(\Gamma\), i. e. \(\dfrac{\partial h}{\partial \mathbf n} = 0\) for any normal \(\mathbf n \) to \(\Gamma\), then \[\frac{\partial H}{\partial \mathbf N} \text{ for any normal } \mathbf N \text{ to } C.\]
Dirichlet and Neumann problems
Example: In \(\C\) (\(w\)-plane), the function \[h(u,v) = v = \Im(w)\] is harmonic, in particular on the horizontal strip \(\Lambda\), \[-\frac{\pi}{2} \le v \le \frac{\pi}{2} .\]
Claim: \(f:z \mapsto \Log z\), maps \(\Omega\) onto the interior of \(\Lambda\).
Dirichlet and Neumann problems
Sketch of mapping
Dirichlet and Neumann problems
\(z \mapsto \Log z\) \(= \ln |z| + i \Arg (z)\)
\(\qquad \quad \;\; \;= \ln \sqrt{x^2+y^2} + i \arctan \left(\dfrac{y}{x}\right)\).
\(H(x,y)= \arctan \left(\dfrac{y}{x}\right)\) on \(\Omega\).