What are harmonic functions and how do I solve the Dirichlet problem on a disk?

**Physical Significance:** - **Steady-state heat:** Temperature distribution in a region with fixed boundary temperatures - **Electrostatics:** Electric potential in a charge-free region - **Fluid flow:** Velocity potential for incompressible, irrotational flow **Mean Value Property:** If $u$ is harmonic in a domain, then for any disk $B_r(z_0)$: $$u(z_0) = \frac{1}{2\pi} \int_0^{2\pi} u(z_0 + re^{i\theta}) \, d\theta$$ The value at the center equals the average over the boundary. This characterises harmonic functions (converse also holds). **Connection to Analytic Functions:** If $f(z) = u(x, y) + iv(x, y)$ is analytic, then both $u$ and $v$ are harmonic. Every harmonic function is the real part of some analytic function (locally). Given $u$, we can find $v$ via the Cauchy-Riemann equations. **Poisson's Integral Formula for the Unit Disk:** The solution to the Dirichlet problem on the unit disk is: $$u(r, \theta) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - \phi) + r^2} \, f(\phi) \, d\phi$$ **Example: $f(\theta) = \sin^2 \theta$ on the unit disk.** Using $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$, the solution is: $$u(r, \theta) = \frac{1}{2} - \frac{r^2}{2} \cos 2\theta$$ This can be verified to satisfy Laplace's equation and the boundary condition (when $r = 1$: $u(1, \theta) = \frac12 - \frac12 \cos 2\theta = \sin^2 \theta$). **Separation of variables approach:** Assume $u(r, \theta) = R(r)\Theta(\theta)$, solve the resulting ODEs, and use Fourier series to match the boundary condition. This gives: $$u(r, \theta) = \frac{a_0}{2} + \sum_{n=1}^{\infty} r^n (a_n \cos n\theta + b_n \sin n\theta)$$ where $a_n, b_n$ are the Fourier coefficients of $f(\theta)$.