📂Partial Differential Equations

Laplace's Equation and Poisson's Equation

Definition¹

• $\ U \in \mathbb{R}^n$ is an open set
• $\ x\in U$
• $u=u(x) : \overline{U} \rightarrow \mathbb{R}^n$

Laplace’s Equation

The partial differential equation below is called Laplace’s equation.

$$\Delta u=0$$

Here, $\Delta$ is the Laplacian. A $u$ that satisfies Laplace’s equation is specifically called a harmonic function.

Poisson’s Equation

The nonhomogeneous Laplace’s equation is called Poisson’s equation.

$$-\Delta u = f$$

Explanation

Laplace’s equation appears in various parts of physics. In most cases, $u$ represents the density of some physical quantity in equilibrium. In equilibrium, when it is said that $V \subset U$, the following equation holds.

$$\int_{\partial V}\mathbf{F} \cdot \boldsymbol{\nu}dS=0$$

$\mathbf{F}$ is the flux density of $u$, $\boldsymbol{\nu}$ is the outward unit normal vector.

The meaning of the equation is that the net flux of $u$ is $0$. For example, suppose there is a space in thermal equilibrium. Then there is no heat entering from the outside into the space, and no heat leaving from the inside to the outside. That is, there is no flow of heat at the boundary of that space. This statement is the same as saying that the net flux is $0$. Applying the Green-Gauss theorem here gives the following equation.

$$0 = \int_{\partial V} \mathbf{F} \cdot \nu dS=\int_{V} \nabla \cdot \mathbf{F} dx \\ \implies \nabla \cdot \mathbf{F}=0$$

Here, let’s assume $\mathbf{F}$ is a value proportional to the gradient $Du$ of $u$. In many cases, it is convenient to assume it in the opposite direction for physical reasons. The second law of thermodynamics (heat always flows from high to low) can be mentioned as an example.

$$\begin{equation} \mathbf{F}=-aDu \label{eq1} \end{equation}$$

In this case, $a>0$.

If $u$ represents the concentration of a chemical substance, temperature, or electrostatic potential, then $\eqref{eq1}$ respectively mean Fick’s law of diffusion, Fourier’s law of heat conduction, Ohm’s law of electrical conduction.

From the above content, the Laplace’s equation is derived.

$$\nabla \cdot \mathbf{F} = \nabla \cdot (-aDu)=-a\Delta u=0 \\ \implies \Delta u = 0$$