In two dimensions, a function $u(x, y)$ is harmonic if and only if it is the real (or imaginary) part of a holomorphic function $f(z) = u(x, y) + iv(x, y)$.

The imaginary part $v$ is the harmonic conjugate of $u$, satisfying the Cauchy-Riemann equations: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$

This correspondence allows complex analysis techniques to solve 2D Laplace problems.

Let $B_t$ be Brownian motion in $\Omega$ and $\tau$ the first exit time from $\Omega$. Then: $u(x) = \mathbb{E}^x[g(B_\tau)]$ solves the Dirichlet problem $\nabla^2 u = 0$ in $\Omega$ with $u = g$ on $\partial\Omega$.

This probabilistic representation shows harmonic functions are expectations of boundary values under Brownian motion, connecting PDEs to stochastic processes.

The eigenvalue problem $-\nabla^2\phi = \lambda\phi$ with $\phi = 0$ on $\partial\Omega$ has:

• Discrete spectrum $0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots$
• Complete orthonormal basis $\{\phi_n\}$ of eigenfunctions
• First eigenvalue $\lambda_1$ satisfies $\lambda_1 = \inf \frac{\int_\Omega |\nabla u|^2}{\int_\Omega u^2}$ over non-zero $u$ with $u = 0$ on $\partial\Omega$

The first eigenvalue measures the "capacity" of the domain—how efficiently it can confine energy.