Laplace and Poisson equations

Separation of variables solves Laplace's equation on special shapes. Here we study harmonic functions on any domain, using integral identities instead of series.

1. Harmonic functions (Def. 1.1) and the fundamental solution (Prop. 1.2).
2. The mean value property (Thm 1.3), the maximum principle (Thm 1.4) and uniqueness (Cor. 1.5).
3. Dirichlet's principle (Prop. 1.6), Green's identities (Lemma 1.7), Green's functions (Thm 1.8) and the Poisson kernel (Example 1.9).
4. Smoothness and Liouville's theorem (Cor. 1.10).

Laplace's equation is $\Delta u = 0$; its $C^2$ solutions are harmonic. Poisson's equation is $-\Delta u = f$. The Dirichlet problem prescribes $u = g$ on $\partial\Omega$; the Neumann problem prescribes $\frac{\partial u}{\partial\nu} = g$. The Laplacian commutes with translations and rotations, so harmonic functions stay harmonic under rigid motions.

The radial harmonic functions on $\mathbb{R}^n\setminus\{0\}$ are $a + b\ln r$ ($n = 2$) and $a + br^{2-n}$ ($n\ge3$). The fundamental solution of $-\Delta$ is

normalised so that the flux of $-\nabla\Phi$ through every sphere about $0$ is $1$.

If $u$ is harmonic in $\Omega$ and $\overline{B(x, r)}\subseteq\Omega$, then

Conversely, a $C^2$ function with the mean value property on all small spheres is harmonic.

Let $u\in C^2(\Omega)\cap C(\overline\Omega)$ be harmonic, $\Omega$ bounded. Weak: $\max_{\overline\Omega}u = \max_{\partial\Omega}u$ (and similarly for the minimum). Strong: if $\Omega$ is connected and $u$ attains its maximum at an interior point, then $u$ is constant.

The Dirichlet problem $-\Delta u = f$ in $\Omega$, $u = g$ on $\partial\Omega$ has at most one solution in $C^2(\Omega)\cap C(\overline\Omega)$, and solutions of Laplace's equation depend continuously on the data: $\max_{\overline\Omega}|u_1 - u_2|\le\max_{\partial\Omega}|g_1 - g_2|$.

Let $\mathcal A = \{w\in C^2(\overline\Omega): w = g\text{ on }\partial\Omega\}$ and $I[w] = \int_\Omega\big(\frac12|\nabla w|^2 - fw\big)\,\mathrm{d}x$. Then $u\in\mathcal A$ solves $-\Delta u = f$ if and only if $I[u] = \min_{w\in\mathcal A}I[w]$.

For $u, v\in C^2(\overline\Omega)$:

Suppose for each $x\in\Omega$ there is a harmonic $\phi^x$ on $\Omega$ with $\phi^x(y) = \Phi(y - x)$ for $y\in\partial\Omega$. The Green's function is $G(x, y) = \Phi(y - x) - \phi^x(y)$. Then a solution $u\in C^2(\overline\Omega)$ of $-\Delta u = f$ in $\Omega$, $u = g$ on $\partial\Omega$, satisfies

Using reflection (image charges), explicit Green's functions exist for half-spaces and balls. The resulting Poisson kernels solve the Dirichlet problem for Laplace's equation:

• unit disc: $u(r, \theta) = \frac1{2\pi}\int_0^{2\pi}\frac{1 - r^2}{1 - 2r\cos(\theta - \phi) + r^2}g(\phi)\,\mathrm{d}\phi$;
• upper half-plane: $u(x, y) = \frac1\pi\int_{-\infty}^\infty\frac{y}{(x - t)^2 + y^2}g(t)\,\mathrm{d}t$. For the half-plane, $G(x, y) = \Phi(y - x) - \Phi(y - x^*)$ with $x^*$ the reflection of $x$.

Harmonic functions are $C^\infty$ (indeed real analytic). Liouville: a bounded harmonic function on all of $\mathbb{R}^n$ is constant.

• Uniqueness: maximum principle (Dirichlet) or energy $\int|\nabla w|^2$ (both, Neumann up to constants).
• Bounds: the max principle bounds $u$ by its boundary values; compare with explicit sub- and supersolutions.
• Formulas: Green's functions by images for half-spaces and balls; Poisson kernels.
• Guess and check: polynomial or separable harmonic functions matching the data, then invoke uniqueness.