Separation of Variable

Harmonic

$-\Delta u=0,$ that is the Laplacian of the function is set to 0.

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title: “Theory of PDE: Separation of Variables and Eigenfunctions of the Laplacian” date: 2026-07-30 tags: [pde, math, laplacian, spherical-harmonics]

Theory of PDE: Separation of Variables and Eigenfunctions of the Laplacian

1. Motivation: The 1D Heat Equation

We begin by recalling the classical separation of variables applied to the 1D heat equation on an interval $[0,\pi ]$ with Dirichlet boundary conditions:

$$\{\begin{array}{ll}{u}_t=u_{xx}& \text{in }[0,\pi ]\\ u(0,t)=0=u(\pi ,t)& \\ u(x,0)=\phi (x)& \end{array}$$

Assuming a separable solution of the form $u(x,t)=T(t)X(x)$, substituting this into the PDE yields:

$$\frac{T_t}{T}=\frac{X_{xx}}{X}=-\lambda$$

This separates into two ordinary differential equations:

1. Time: $T_t=-\lambda T⟹T(t)=C{e}^{-\lambda t}$
2. Space: $-X_{xx}+\lambda X=0$

Subject to the boundary conditions $X(0)=X(\pi )=0$, the spatial equation forms an eigenvalue problem. The non-trivial solutions occur when $\sqrt{\lambda }=k\in \mathbb{N}$, yielding the eigenfunctions $X_k(x)=\sin (kx)$. By superposition, the general solution is:

$$u(x,t)=\sum _{k=1}^{\infty }{C}_k{e}^{-k^{2}t}\sin (kx)$$

The coefficients $C_k$ are determined by the initial condition $\phi (x)={\sum }_{k=1}^{\infty }{C}_k\sin (kx)$, which converges provided $\phi \in L^2([0,\pi ])$ (i.e., ${\int }_0^{\pi }{\phi }^{2}dx<\infty$).

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2. The General Heat Equation and the Dirichlet Eigenvalue Problem

Generalizing to an open domain $\Omega \subset {\mathbb{R}}^n$, the heat equation reads:

$$\{\begin{array}{ll}{u}_t=\Delta u& \text{in }\Omega \\ u=0& \text{at }\partial \Omega \end{array}$$

Using the same ansatz $u(x,t)=T(t)X(x)$, we obtain:

$$\frac{T_t}{T}=\frac{\Delta X}{X}=-\lambda$$

This leaves us with the Dirichlet Eigenvalue Problem for the Laplacian:

$$\{\begin{array}{ll}-\Delta X=\lambda X& \text{in }\Omega \\ X=0& \text{on }\partial \Omega \end{array}$$

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3. Spatial Separation in Spherical Coordinates

When the domain is a ball $B\subset {\mathbb{R}}^n$, it is natural to express the Laplacian in spherical coordinates. For $x=rz$ where $r=|x|$ and $z\in S^{n-1}$, the Laplacian decomposes into radial and angular components:

$$\Delta =\frac{1}{r^{n-1}}\frac{\partial }{\partial r}\left(r^{n-1}\frac{\partial }{\partial r}\right)+\frac{1}{r^2}{\Delta }_{S^{n-1}}$$

where ${\Delta }_{S^{n-1}}$ is the Laplace-Beltrami operator on the sphere $S^{n-1}$.

We perform a second separation of variables on the spatial eigenfunction: $X(r,z)=R(r)\Theta (z)$. Substituting this into $-\Delta X=\lambda X$ gives:

$$-\left(\frac{1}{r^{n-1}}\frac{\partial }{\partial r}\left(r^{n-1}{R}_r\right)\Theta +\frac{1}{r^2}R{\Delta }_{S^{n-1}}\Theta \right)=\lambda R\Theta$$

Dividing by $R\Theta$ and isolating the angular and radial terms yields:

$$\lambda r^2+\frac{r^2}{R}\frac{1}{r^{n-1}}(r^{n-1}{R}_r{)}_r=-\frac{{\Delta }_{S^{n-1}}\Theta }{\Theta }=\mu$$

This successfully decouples the angular dependence, producing an eigenvalue problem purely on the manifold $S^{n-1}$:

$$-{\Delta }_{S^{n-1}}\Theta =\mu \Theta \text{on }{S}^{n-1}$$

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4. Eigenfunctions of the Spherical Laplacian (Spherical Harmonics)

To solve $-{\Delta }_{S^{n-1}}\Theta =\mu \Theta$, we introduce an algebraic approach using polynomials in ${\mathbb{R}}^n$.

Homogeneous Polynomials

A polynomial $p:{\mathbb{R}}^n\to \mathbb{C}$ is homogeneous of degree $k$ if for all $\lambda >0$: $p(\lambda x)={\lambda }^{k}p(x)$ We denote the space of such polynomials as $P_k({\mathbb{R}}^n,\mathbb{C})$.

Harmonic Polynomials

The space of degree $k$ harmonic homogeneous polynomials is defined as: $H_k({\mathbb{R}}^n,\mathbb{C}):=\{p\in P_k({\mathbb{R}}^n,\mathbb{C})\mid -\Delta p=0\}$

Spherical Harmonics

The space of spherical harmonics of degree $k$ is the restriction of $H_k({\mathbb{R}}^n,\mathbb{C})$ to the unit sphere: $H_k(S^{n-1},\mathbb{C}):=\{p{|}_{S^{n-1}}\mid p\in H_k({\mathbb{R}}^n,\mathbb{C})\}$

Restriction Isomorphism

The restriction map $p\mapsto p{|}_{S^{n-1}}$ is a linear isomorphism between $H_k({\mathbb{R}}^n,\mathbb{C})$ and $H_k(S^{n-1},\mathbb{C})$.

Eigenvalues of the Laplace-Beltrami Operator

The spherical harmonics are exactly the eigenfunctions of ${\Delta }_{S^{n-1}}$: $H_k(S^{n-1},\mathbb{C})=\{\Theta \in C^{\infty }(S^{n-1})\mid -{\Delta }_{S^{n-1}}\Theta =k(k+n-2)\Theta \}$

Proof sketch: Evaluate the Laplacian of $p(x)=p(rz)=r^{k}p(z)$ in spherical coordinates. Since $\Delta p=0$, applying the decomposed Laplacian formula directly relates the radial derivatives of $r^k$ to the angular operator ${\Delta }_{S^{n-1}}$, yielding the eigenvalue $\mu =k(k+n-2)$.

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5. Orthogonality of Spherical Harmonics

Orthogonality

The spaces $H_k(S^{n-1},\mathbb{C})$ and $H_{\ell }(S^{n-1},\mathbb{C})$ are pairwise orthogonal in $L^2(S^{n-1})$ when $k\ne \ell$. That is, for $P\in H_k$ and $Q\in H_{\ell }$, ${\int }_{S^{n-1}}P\bar{Q}d\sigma =0$.

Proof Outline

Let $p(x):=|x{|}^{k}P\left(\frac{x}{|x|}\right)$ and $q(x):=|x{|}^{\ell }Q\left(\frac{x}{|x|}\right)$. By definition, $p$ and $q$ are harmonic in ${\mathbb{R}}^n$.

By Euler’s homogeneous function theorem, the normal derivative on the boundary of the unit ball $B^n$ is given by: $Dp\cdot \frac{x}{|x|}=kP\text{at }{S}^{n-1}$

Using Green’s second identity on the unit ball $B^n$: ${\int }_{B^n}(\bar{q}\Delta p-p\Delta \bar{q})dx={\int }_{\partial B^n=S^{n-1}}\left(\bar{q}\frac{\partial p}{\partial n}-p\frac{\partial \bar{q}}{\partial n}\right)d\sigma$

Since $p,q$ are harmonic, $\Delta p=\Delta q=0$, so the volume integral vanishes. On the boundary $S^{n-1}$, we substitute the normal derivatives: $0={\int }_{S^{n-1}}(\bar{Q}(kP)-P(\ell \bar{Q}))d\sigma =(k-\ell ){\int }_{S^{n-1}}P\bar{Q}d\sigma$

Assuming $k\ne \ell$, it follows immediately that ${\int }_{S^{n-1}}P\bar{Q}d\sigma =0$.