The Laplacian and Its Physical Applications

1Introduction

This lecture explains how the Laplacian characterizes deviation from a local average and serves as a central operator in the mathematical descriptions of diffusion, waves, and potentials.

2Terminology and Definition

Let $\mathrm{\Omega }\subset R^n$\Omega\subset\mathbb R^n be open. For a scalar field $f\in C^2\left(\mathrm{\Omega }\right)$f\in C^2(\Omega), the Laplacian is the second-order linear differential operator

In Cartesian coordinates on $R^n$\mathbb R^n,

This lecture concerns the scalar Laplacian in Euclidean space. The preceding lecture used the divergence theorem to show that the volume integral of $\mathrm{\Delta }u$\Delta u equals the outward boundary flux of $\nabla u$\nabla u. Here, the local significance of this relation is explained through local averages.

3Strategy

The Laplacian supplies the leading term in the difference between the value at a point and the average over a small sphere surrounding that point. We first derive this relation from a Taylor expansion and then connect it to the heat, wave, Laplace, and Poisson equations.

4Deviation from a Local Average

Let $f\in C^2\left(\mathrm{\Omega }\right)$f\in C^2(\Omega) and $x\in \mathrm{\Omega }$x\in\Omega, and suppose that the closed ball of radius $r$r centered at $x$x lies in $\mathrm{\Omega }$\Omega. Define the spherical average by

A Taylor expansion as $r\to 0$r\to0 gives

Here $o\left(r^2\right)$o(r^2) denotes a remainder satisfying $o\left(r^2\right)/r^2\to 0$o(r^2)/r^2\to0 as $r\to 0$r\to0. The linear terms and the off-diagonal entries of the Hessian have zero spherical average by symmetry. The sum of its diagonal entries remains and equals the Laplacian. Consequently, if $\mathrm{\Delta }f\left(x\right)>0$\Delta f(x)>0, the leading term dominates the remainder and the surrounding average exceeds the central value for every sufficiently small $r$r; if $\mathrm{\Delta }f\left(x\right)<0$\Delta f(x)<0, the inequality is reversed. The condition $\mathrm{\Delta }f\left(x\right)=0$\Delta f(x)=0 at one point alone does not imply equality with an average over a ball of finite radius.

In one dimension, the Laplacian is the second derivative and characterizes the local concavity or convexity of a graph. In higher dimensions, it sums second-order changes in mutually orthogonal directions.

Suppose $\mathrm{\Delta }f=0$\Delta f=0 on a neighborhood of $\stackrel{\_}{B_R\left(x\right)}$\overline{B_R(x)}. Differentiating the spherical average after expressing it as an integral over the unit sphere, and then applying the divergence theorem, gives, for $0<r<R$0<r<R,

Thus $M_{r}f\left(x\right)$M_r f(x) is independent of $r$r. Since $M_{r}f\left(x\right)\to f\left(x\right)$M_r f(x)\to f(x) as $r\to 0$r\to0,

for every $0<r<R$0<r<R. This identity is the mean-value property.

5Principal Equations

We consider the simplest homogeneous, isotropic, constant-coefficient classical models and solutions with sufficient regularity: $C^2$C^2 in space and $C^1$C^1 in time for the heat equation, $C^2$C^2 in both space and time for the wave equation, and $C^2$C^2 in space for the Laplace and Poisson equations.

For thermal diffusivity $\kappa >0$\kappa>0, the heat equation is

It describes the evolution of temperature toward its surrounding average. The Laplace equation

describes an equilibrium without an interior source, and its solutions are harmonic functions.

6Examples of Harmonic Functions

For $u(x,y)=x^2-y^2$u(x,y)=x^2-y^2, one has $u_{xx}=2$u_{xx}=2 and $u_{yy}=-2$u_{yy}=-2, hence $\mathrm{\Delta }u=0$\Delta u=0. Thus $u$u is harmonic and satisfies the mean-value property on every ball contained in its domain.

Another example is

which is harmonic away from the origin. Because it has a singularity at the origin, it must not be described as harmonic on the entire plane. This example leads to Green functions and fundamental solutions.

7Dependence of the Formula on Coordinates

In two-dimensional Cartesian coordinates, $\mathrm{\Delta }f=f_{xx}+f_{yy}$\Delta f=f_{xx}+f_{yy}. In polar coordinates with $r>0$r>0,

This formula follows by applying the chain rule to $x=r\cos \theta$x=r\cos\theta and $y=r\sin \theta$y=r\sin\theta and simplifying $f_{xx}+f_{yy}$f_{xx}+f_{yy}. Polar coordinates are singular at $r=0$r=0, so this expression cannot be substituted there directly. The Euclidean operator itself is unchanged, but its coordinate expression changes; terms such as $1/r$1/r arise from the coordinate transformation.

8Dimensional Check

In the heat equation, suppose that $u$u is measured in $\mathrm{K}$\mathrm K and length in $\mathrm{m}$\mathrm m. Then $\mathrm{\Delta }u$\Delta u has units $\mathrm{K}/m^2$\mathrm{K/m^2}. If $\kappa$\kappa has units $m^2/\mathrm{s}$\mathrm{m^2/s}, then $\kappa \mathrm{\Delta }u$\kappa\Delta u has units $\mathrm{K}/\mathrm{s}$\mathrm{K/s}, which agrees with $u_t$u_t.

9Common Confusions

The scalar Laplacian acts on scalar fields. For a $C^2$C^2 vector field $\mathit{F}$\boldsymbol F in Cartesian coordinates on $R^3$\mathbb R^3, the vector Laplacian may be defined componentwise, and

Expanding the right side componentwise cancels the mixed partial derivatives and leaves $\mathrm{\Delta }{F}_i$\Delta F_i in each component. In curvilinear coordinates, merely differentiating the displayed components twice omits the variation of the basis and therefore does not produce the same operator.

10Scope of Validity

The discussion above concerns the Euclidean Laplacian and homogeneous, isotropic, constant-coefficient models. In an inhomogeneous or anisotropic medium, a variable-coefficient operator such as $\nabla ·\left(A\right(x)\nabla u)$\nabla\cdot(A(x)\nabla u) may appear. On a general Riemannian manifold, the metric determines the Laplace--Beltrami operator. Moreover, the mean-value property follows from harmonicity on a region containing the ball, not merely from the equation $\mathrm{\Delta }f=0$\Delta f=0 at its center.

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