Laplace's Equation

The Equation

$\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$

In 3D: $\nabla^2 \phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0$

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Physical Context

• Steady-state temperature distribution
• Electrostatic potential (no charges)
• Gravitational potential (no masses)
• Incompressible, irrotational fluid flow

Key property: Solutions are harmonic functions — smooth, with no local maxima/minima (maximum principle).

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Separation of Variables

Problem: Find $\phi(x, y)$ in rectangle $0 < x < a$, $0 < y < b$ with:

• $\phi(0, y) = 0$ (left edge)
• $\phi(x, 0) = 0$ (bottom edge)
• $\phi(x, b) = 0$ (top edge)
• $\phi(a, y) = f(y)$ (right edge)

Ansatz: $\phi(x, y) = X(x)Y(y)$

Substituting into Laplace equation:

$X''Y + XY'' = 0$

$\frac{X''}{X} = -\frac{Y''}{Y} = k$

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Solution

For homogeneous Dirichlet conditions on $y$ edges:

The separation constant must be negative: $k = -\gamma^2 < 0$.

$Y$ equation: $Y'' + \gamma^2 Y = 0$

$Y(y) = A\sin(\gamma y) + B\cos(\gamma y)$

Boundary condition $Y(0) = 0 \Rightarrow B = 0$.

$Y(b) = 0 \Rightarrow \sin(\gamma b) = 0$, so $\gamma = \frac{n\pi}{b}$ for $n = 1, 2, 3, \ldots$

$X$ equation: $X'' - \gamma^2 X = 0$

$X(x) = C\sinh(\gamma x) + D\cosh(\gamma x)$

Boundary condition $X(0) = 0 \Rightarrow D = 0$.

Combined solution:

$\phi(x, y) = \sum_{n=1}^{\infty} C_n \sin\left(\frac{n\pi y}{b}\right) \sinh\left(\frac{n\pi x}{b}\right)$

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Applying the Inhomogeneous Boundary Condition

At $x = a$: $\phi(a, y) = f(y)$

This is a Fourier sine series, so:

$f(y) = \sum_{n=1}^{\infty} C_n \sin\left(\frac{n\pi y}{b}\right) \sinh\left(\frac{n\pi a}{b}\right)$

Fourier coefficients:

$C_n = \frac{2}{b\sinh(n\pi a/b)} \int_0^b f(y)\sin\left(\frac{n\pi y}{b}\right) dy$

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Example: Constant $f(y) = 1$

$C_n = \frac{2}{b\sinh(n\pi a/b)} \int_0^b \sin\left(\frac{n\pi y}{b}\right) dy$

$= \frac{2}{b\sinh(n\pi a/b)} \cdot \frac{b}{n\pi}[1 - (-1)^n]$

$= \frac{2}{n\pi\sinh(n\pi a/b)} \times \begin{cases} 2 & n \text{ odd} \\ 0 & n \text{ even} \end{cases}$

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Maximum Principle

Statement: A harmonic function on a bounded domain attains its maximum and minimum on the boundary.

Physical interpretation: Steady-state temperature cannot be hotter than the hottest boundary point.

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Uniqueness

For given boundary conditions, Laplace’s equation has a unique solution.

Proof: If $\phi_1$ and $\phi_2$ are solutions, then $\psi = \phi_1 - \phi_2$ is harmonic with zero boundary conditions. By maximum principle, $\psi = 0$.

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Why Laplace’s Equation Matters

• Describes equilibrium in many physical systems
• Solved analytically using separation of variables
• Basis for understanding more complex PDEs
• Important in electrostatics, fluid dynamics, and heat transfer