# MATH3401

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 $$\Omega$$.