Laplace equation and boundary value problems

Intuition [Beginner]

In a region with no electric charges, the electric potential $V$ satisfies Laplace's equation: the sum of its second spatial derivatives is zero at every point. The potential is a smooth surface with no sources or sinks — it cannot have a local maximum or minimum anywhere inside the charge-free region.

A direct consequence: the value of $V$ at any interior point equals the average of $V$ over a small sphere centred on that point. If $V$ were larger than all its neighbours somewhere, the average would fall short — a contradiction. This mean value property rules out interior peaks and valleys.

To get a unique solution you must fix boundary conditions — the values of $V$ on the surface enclosing the region. This is the Dirichlet problem: find a function satisfying Laplace's equation inside that matches given values on the boundary. One technique is the method of images: place a fake "image" charge outside the region so that the combined potential of real and image charges satisfies the boundary condition. Uniqueness then guarantees the result is correct.

Visual [Beginner]

Imagine a rubber sheet stretched over a frame. Clamp each edge at a chosen height. The sheet settles into the shape of minimum energy — no interior bumps or sags. The height of the sheet satisfies Laplace's equation; the clamped edges are the boundary conditions.

For electrostatics, replace "rubber-sheet height" with "electric potential" and "clamped edges" with "voltages on conducting surfaces." The potential between conductors is as smooth as the rubber sheet. It has no local peaks or valleys inside.

Worked example [Beginner]

A point charge $q$ sits at height $d$ above a flat conducting plane held at $V=0$. The plane is grounded — charge redistributes freely to keep the potential at zero.

Step 1: set up the image. Place a fake charge $-q$ at the mirror position, a depth $d$ below the plane. On the plane ($z=0$), every point is equidistant from both charges, so their potentials cancel and $V=0$.

Step 2: write the potential above the plane. For $z>0$,

$$V=\frac{q}{4\pi {ϵ}_0}\left(\frac{1}{r_{+}}-\frac{1}{r_{-}}\right),$$

where $r_{+}=\sqrt{x^2+y^2+(z-d{)}^2}$ is the distance to the real charge and $r_{-}=\sqrt{x^2+y^2+(z+d{)}^2}$ is the distance to the image.

Step 3: verify. Both terms are point-charge potentials, so each satisfies Laplace's equation on its own. The boundary condition $V=0$ at $z=0$ holds by construction. Uniqueness says this is the only solution.

Step 4: force. The real charge feels the Coulomb force from the image: $F=q^2/(16\pi {ϵ}_0{d}^2)$, directed toward the plane. The grounded plane attracts the charge as though an opposite charge sat behind it.

Check your understanding [Beginner]

Formal definition [Intermediate+]

In a region $\mathcal{V}$ where the charge density $\rho$ vanishes, Gauss's law 10.01.01 pending gives $\nabla \cdot \mathbf{E}=0$. Writing $\mathbf{E}=-\nabla V$ and requiring the curl to vanish (electrostatics), the potential satisfies Laplace's equation

$${\nabla }^{2}V=\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}+\frac{{\partial }^{2}V}{\partial z^2}=0.$$

In spherical coordinates $(r,\theta ,\phi )$:

$${\nabla }^{2}V=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial V}{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial V}{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^{2}V}{\partial {\phi }^2}=0.$$

In cylindrical coordinates $(s,\phi ,z)$:

$${\nabla }^{2}V=\frac{1}{s}\frac{\partial }{\partial s}\left(s\frac{\partial V}{\partial s}\right)+\frac{1}{s^2}\frac{{\partial }^{2}V}{\partial {\phi }^2}+\frac{{\partial }^{2}V}{\partial z^2}=0.$$

A function satisfying ${\nabla }^{2}V=0$ on an open set is called harmonic. Harmonic functions satisfy two fundamental properties:

1. Mean value property. For any point ${\mathbf{r}}_0$ in the domain, $V({\mathbf{r}}_0)$ equals the average of $V$ over any sphere $S_R$ centred at ${\mathbf{r}}_0$ and contained in the domain:

$$V({\mathbf{r}}_0)=\frac{1}{4\pi R^2}{\oint }_{S_R}Vda.$$

2. Maximum principle. A non-constant harmonic function on a connected open set attains its maximum and minimum values only on the boundary.

Boundary conditions

Three standard types:

• Dirichlet: $V$ is specified on $S$. "Fix the voltages on the conductors."
• Neumann: $\partial V/\partial n$ (the outward normal derivative) is specified on $S$. "Fix the surface charge density on the conductors."
• Mixed (Robin): Dirichlet on part of $S$, Neumann on the rest.

For Neumann conditions the solution is unique up to an additive constant (the potential is determined only to within a gauge). For Dirichlet conditions the solution is unique. Both statements are proved below.

Separation of variables

The standard constructive method for solving Laplace's equation in regions whose boundaries align with coordinate surfaces. Write $V$ as a product of single-variable functions, substitute into Laplace's equation, and obtain ODEs for each factor.

Cartesian. $V(x,y,z)=X(x)Y(y)Z(z)$. Substitution gives

$$\frac{X^{\prime \prime }}{X}+\frac{Y^{\prime \prime }}{Y}+\frac{Z^{\prime \prime }}{Z}=0.$$

Each ratio must be a constant (say $k_x^2$, $k_y^2$, $-(k_x^2+k_y^2)$), yielding exponential or trigonometric solutions. Boundary conditions determine the separation constants and the expansion coefficients.

Spherical. For azimuthal symmetry ($V$ independent of $\phi$), write $V(r,\theta )=R(r)\Theta (\theta )$. Separation gives

$$\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)=-\frac{1}{\Theta \sin \theta }\frac{d}{d\theta }\left(\sin \theta \frac{d\Theta }{d\theta }\right)=\ell (\ell +1),$$

where $\ell$ is a non-negative integer separation constant. The angular equation produces Legendre's equation, with regular solutions $P_{\ell }(\cos \theta )$ (Legendre polynomials). The radial equation gives $R(r)=A{r}^{\ell }+B{r}^{-(\ell +1)}$. The general azimuthally symmetric solution is

$$V(r,\theta )=\sum _{\ell =0}^{\infty }\left(A_{\ell }{r}^{\ell }+\frac{B_{\ell }}{r^{\ell +1}}\right)P_{\ell }(\cos \theta ).$$

The full azimuthal dependence introduces associated Legendre functions $P_{\ell }^m(\cos \theta )$ and spherical harmonics $Y_{\ell }^m(\theta ,\phi )$, treated systematically in Jackson Ch. 3.

Cylindrical. $V(s,\phi ,z)=S(s)\Phi (\phi )Z(z)$. The radial equation becomes Bessel's equation, with solutions $J_m(ks)$ and $Y_m(ks)$ (Bessel functions of the first and second kind). This arises in problems with cylindrical conducting boundaries.

Method of images

The idea: replace a complicated boundary-value problem with a simpler one involving fictitious charges. The method works when the boundary geometry admits a finite (or tractably infinite) set of image charges that satisfy the boundary condition exactly.

Plane. Treated in the Beginner tier: a point charge $q$ at distance $d$ from a grounded plane is mirrored by $-q$ at the mirror point.

Sphere. A point charge $q$ at distance $a$ from the centre of a grounded conducting sphere of radius $R$ (with $a>R$) requires an image charge $q^{\prime }=-(R/a)q$ at distance $b=R^2/a$ from the centre, along the radial line from the centre to $q$. This preserves the spherical boundary at $V=0$. For a sphere held at non-zero potential $V_0$, add a second image charge $4\pi {ϵ}_{0}R{V}_0$ at the centre.

Cylinder. An infinite grounded conducting cylinder of radius $R$ with a parallel line charge $\lambda$ at distance $d$ from the axis requires an image line charge $-\lambda$ at distance $R^2/d$ from the axis. The 2D cross-section is conformally equivalent to the plane problem.

Multipole expansion

Far from a localised charge distribution, the potential can be expanded in inverse powers of $r$:

$$V(\mathbf{r})=\frac{1}{4\pi {ϵ}_0}\left[\frac{Q}{r}+\frac{\mathbf{p}\cdot \hat{\mathbf{r}}}{r^2}+\frac{1}{2r^3}\sum _{i,j}{Q}_{ij}(3{\hat{r}}_i{\hat{r}}_j-{\delta }_{ij})+\cdots \right],$$

where $Q=\int \rho d\tau$ is the total charge (monopole moment), $\mathbf{p}=\int {\mathbf{r}}^{\prime }\rho ({\mathbf{r}}^{\prime })d{\tau }^{\prime }$ is the dipole moment, and $Q_{ij}=\int r_i^{\prime }{r}_j^{\prime }\rho ({\mathbf{r}}^{\prime })d{\tau }^{\prime }$ is the quadrupole tensor. Each successive term falls off faster; at large $r$ the leading non-vanishing term dominates.

For a neutral distribution ($Q=0$), the dipole term dominates. If $\mathbf{p}=0$ also, the quadrupole dominates. The multipole expansion is the Taylor expansion of $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$ about ${\mathbf{r}}^{\prime }=0$; it converges only outside the smallest sphere enclosing all the charge.

Counterexamples to common slips

• Laplace's equation does not hold where charges exist. Inside a region with $\rho \ne 0$ the governing equation is Poisson's equation ${\nabla }^{2}V=-\rho /{ϵ}_0$. Applying Laplace methods to a charge-containing region without accounting for the source term produces wrong results.
• The image-charge potential is valid only inside the region of interest. Outside the region the image potential is physically meaningless. For the point charge above a grounded plane, the image formula holds for $z>0$; below the plane the field is zero (inside the conductor) and the image formula does not apply.
• Separation of variables requires the boundary to align with a coordinate surface. Rectangular boundaries require Cartesian coordinates, spherical boundaries require spherical coordinates. A boundary that does not match any standard coordinate system is not amenable to elementary separation of variables — numerical methods or Green's functions (Master tier) are needed.
• The multipole expansion converges only outside the smallest sphere containing all the charge. Using it inside the charge distribution gives a divergent or meaningless series.
• The Neumann problem determines $V$ only up to a constant. Specifying $\partial V/\partial n$ on the boundary fixes the shape of $V$ but not its absolute level. This is not an error — it reflects the gauge freedom of the potential.

Key theorem with proof [Intermediate+]

Theorem (First uniqueness theorem — Dirichlet). The solution to Laplace's equation in a region $\mathcal{V}$, subject to specified values of $V$ on the boundary $S=\partial \mathcal{V}$, is unique.

Proof. Suppose $V_1$ and $V_2$ both satisfy ${\nabla }^{2}V=0$ in $\mathcal{V}$ and $V_1=V_2$ on $S$. Define $\varphi :=V_1-V_2$. Then ${\nabla }^2\varphi =0$ in $\mathcal{V}$ (linearity of the Laplacian) and $\varphi =0$ on $S$.

Apply Green's first identity with $\psi =\varphi$:

$${\int }_{\mathcal{V}}\left(\varphi {\nabla }^2\varphi +|\nabla \varphi {|}^2\right)d\tau ={\oint }_S\varphi \frac{\partial \varphi }{\partial n}da.$$

The left-hand volume integral: ${\nabla }^2\varphi =0$ kills the first term, leaving ${\int }_{\mathcal{V}}|\nabla \varphi {|}^{2}d\tau$. The right-hand surface integral: $\varphi =0$ on $S$ kills it entirely. So

$${\int }_{\mathcal{V}}|\nabla \varphi {|}^{2}d\tau =0.$$

The integrand $|\nabla \varphi {|}^2$ is non-negative everywhere. The only way its integral vanishes is $\nabla \varphi =0$ at every point, which means $\varphi$ is constant. Since $\varphi =0$ on $S$, that constant is zero. So $\varphi \equiv 0$ and $V_1=V_2$ everywhere in $\mathcal{V}$. $\square$

Corollary (Second uniqueness theorem — Neumann). If $\partial V/\partial n$ is specified on $S$ and ${\nabla }^{2}V=0$ in $\mathcal{V}$, then the solution is unique up to an additive constant.

Proof. The same difference function $\varphi =V_1-V_2$ satisfies $\partial \varphi /\partial n=0$ on $S$ (both solutions share the same normal derivative). Green's first identity again gives ${\int }_{\mathcal{V}}|\nabla \varphi {|}^{2}d\tau =0$ — the surface integral now vanishes because $\partial \varphi /\partial n=0$, not because $\varphi =0$. So $\nabla \varphi =0$ and $\varphi =\text{const}$. Neumann data alone cannot fix this constant; it is the gauge freedom of the potential.

Mixed boundary conditions (Dirichlet on $S_D$, Neumann on $S_N$, with $S=S_D\cup S_N$) are handled the same way: the surface integral splits into two pieces, each vanishing on its respective part, and uniqueness follows.

Worked example: separation of variables in a rectangular pipe

An infinitely long rectangular pipe has cross-section $0\le x\le a$, $0\le y\le b$. Three walls are grounded ($V=0$); the wall at $x=a$ is held at $V_0$. Find $V(x,y)$ inside.

Laplace's equation in 2D: ${\partial }^{2}V/\partial x^2+{\partial }^{2}V/\partial y^2=0$. The boundary conditions are $V(0,y)=0$, $V(x,0)=0$, $V(x,b)=0$, $V(a,y)=V_0$.

Separate: $V(x,y)=X(x)Y(y)$. Substituting and dividing by $XY$:

$$\frac{X^{\prime \prime }}{X}=-\frac{Y^{\prime \prime }}{Y}=-k^2,$$

where $k^2>0$ is the separation constant (sign chosen to give oscillatory solutions in $y$, needed to satisfy the homogeneous boundary conditions $Y(0)=Y(b)=0$). The $Y$-equation $Y^{\prime \prime }+k^{2}Y=0$ with $Y(0)=Y(b)=0$ gives $Y_n(y)=\sin (n\pi y/b)$ with $k_n=n\pi /b$ for $n=1,2,3,\dots$.

The $X$-equation for each $n$ is $X_n^{\prime \prime }-k_n^2{X}_n=0$ with $X_n(0)=0$, giving $X_n(x)=\sinh (k_{n}x)$.

The general solution satisfying the three grounded walls is

$$V(x,y)=\sum _{n=1}^{\infty }{C}_n\sinh \left(\frac{n\pi x}{b}\right)\sin \left(\frac{n\pi y}{b}\right).$$

Apply the fourth boundary condition $V(a,y)=V_0$:

$$V_0=\sum _{n=1}^{\infty }{C}_n\sinh \left(\frac{n\pi a}{b}\right)\sin \left(\frac{n\pi y}{b}\right).$$

This is a Fourier sine series for the constant $V_0$ on $[0,b]$. Multiplying by $\sin (m\pi y/b)$ and integrating over $[0,b]$ gives the coefficients:

$$C_n=\frac{2}{b\sinh (n\pi a/b)}{\int }_0^b{V}_0\sin \left(\frac{n\pi y}{b}\right)dy=\frac{2V_0(1-\cos n\pi )}{n\pi \sinh (n\pi a/b)}.$$

For even $n$, $1-\cos n\pi =0$. For odd $n$, $1-\cos n\pi =2$. The final solution is

$$\boxed{V(x,y)=\frac{4V_0}{\pi }\sum _{\begin{array}{c}n=1,3,5,\dots \end{array}}^{\infty }\frac{1}{n}\frac{\sinh (n\pi x/b)}{\sinh (n\pi a/b)}\sin \left(\frac{n\pi y}{b}\right)}$$

Each term satisfies Laplace's equation individually; the sum converges pointwise inside the pipe. The first term ($n=1$) dominates far from the $x=a$ wall, giving the qualitative shape of the potential.

Exercises [Intermediate+]

Green's functions for the Laplacian [Master]

The Green's function $G(\mathbf{r},{\mathbf{r}}^{\prime })$ for the Laplacian in a region $\mathcal{V}$ satisfies

$${\nabla }^{2}G(\mathbf{r},{\mathbf{r}}^{\prime })=-{\delta }^{(3)}(\mathbf{r}-{\mathbf{r}}^{\prime }),$$

with appropriate boundary conditions on $S=\partial \mathcal{V}$. Physically, $G$ is the potential at $\mathbf{r}$ due to a unit point charge at ${\mathbf{r}}^{\prime }$ in the presence of grounded or otherwise conditioned boundaries. In free space ($\mathcal{V}={\mathbb{R}}^3$, no boundaries):

$$G_0(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{1}{4\pi |\mathbf{r}-{\mathbf{r}}^{\prime }|}.$$

Dirichlet Green's function

$G_D$ satisfies $G_D(\mathbf{r},{\mathbf{r}}^{\prime })=0$ for $\mathbf{r}\in S$ (all ${\mathbf{r}}^{\prime }$). Given Dirichlet data $V=f$ on $S$, the solution to Laplace's equation in $\mathcal{V}$ is

$$V(\mathbf{r})=-{\oint }_{S}f({\mathbf{r}}^{\prime })\frac{\partial G_D}{\partial n^{\prime }}(\mathbf{r},{\mathbf{r}}^{\prime })d{a}^{\prime },$$

where $\partial /\partial n^{\prime }$ is the outward normal derivative with respect to the source coordinate ${\mathbf{r}}^{\prime }$. Once $G_D$ is known for a given domain, any set of boundary values is handled by a single surface integral. Constructing $G_D$ is the hard part; evaluating the integral is mechanical.

Neumann Green's function

$G_N$ satisfies $\partial G_N/\partial n^{\prime }=-4\pi /\mathcal{A}$ on $S$ (where $\mathcal{A}$ is the area of $S$), a condition that enforces the Gauss-law constraint for the Neumann problem. The solution with Neumann data $\partial V/\partial n=g$ on $S$ is

$$V(\mathbf{r})=⟨V{⟩}_S+{\oint }_S{G}_N(\mathbf{r},{\mathbf{r}}^{\prime })g({\mathbf{r}}^{\prime })d{a}^{\prime },$$

where $⟨V{⟩}_S$ is the average of $V$ over $S$ — the undetermined constant inherent to the Neumann problem.

Symmetry and reciprocity

Both $G_D$ and $G_N$ are symmetric: $G(\mathbf{r},{\mathbf{r}}^{\prime })=G({\mathbf{r}}^{\prime },\mathbf{r})$. The proof applies Green's second identity to $G(\mathbf{r},\cdot )$ and $G({\mathbf{r}}^{\prime },\cdot )$ on $\mathcal{V}$ minus two small balls around $\mathbf{r}$ and ${\mathbf{r}}^{\prime }$, uses the defining equation ${\nabla }^{\prime 2}G=-\delta$, and notes that the boundary terms vanish by the boundary conditions on $G$. Physically: the potential at $\mathbf{r}$ due to a unit charge at ${\mathbf{r}}^{\prime }$ equals the potential at ${\mathbf{r}}^{\prime }$ due to a unit charge at $\mathbf{r}$.

Constructing Green's functions by images

For the upper half-space ($z>0$, grounded plane at $z=0$):

$$G_D(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{1}{4\pi }\left(\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}-\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime \prime }|}\right),$$

where ${\mathbf{r}}^{\prime \prime }=(x^{\prime },y^{\prime },-z^{\prime })$ is the reflection of ${\mathbf{r}}^{\prime }$ across $z=0$. For the grounded sphere of radius $R$ centred at the origin:

$$G_D(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{1}{4\pi }\left(\frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}-\frac{R/r^{\prime }}{|\mathbf{r}-(R^2/r^{\prime 2}){\mathbf{r}}^{\prime }|}\right),$$

where $r^{\prime }=|{\mathbf{r}}^{\prime }|$. For general domains lacking this kind of symmetry, $G$ must be found by separation of variables, eigenfunction expansion, or numerical methods.

Eigenfunction expansion

On a bounded domain $\mathcal{V}$ with Dirichlet boundary conditions, the Laplacian has a discrete spectrum $0<{\lambda }_1\le {\lambda }_2\le \cdots$ with eigenfunctions ${\psi }_n$ satisfying ${\nabla }^2{\psi }_n=-{\lambda }_n{\psi }_n$ and ${\psi }_n{|}_S=0$. The Dirichlet Green's function expands as

$$G_D(\mathbf{r},{\mathbf{r}}^{\prime })=\sum _{n=1}^{\infty }\frac{{\psi }_n(\mathbf{r}){\psi }_n({\mathbf{r}}^{\prime })}{{\lambda }_n}.$$

This representation makes the symmetry $G_D(\mathbf{r},{\mathbf{r}}^{\prime })=G_D({\mathbf{r}}^{\prime },\mathbf{r})$ manifest and shows that $G_D$ is a positive-definite kernel. It converges for $\mathbf{r}\ne {\mathbf{r}}^{\prime }$; at coincidence the sum diverges logarithmically (in 2D) or as $1/|\mathbf{r}-{\mathbf{r}}^{\prime }|$ (in 3D), reflecting the point-charge singularity.

Conformal mapping in two dimensions [Master]

In two dimensions, Laplace's equation ${\partial }^{2}V/\partial x^2+{\partial }^{2}V/\partial y^2=0$ is the real part of the Cauchy-Riemann equations. If $f(z)=u(x,y)+iv(x,y)$ is holomorphic, both $u$ and $v$ are harmonic. The conjugate pair $(u,v)$ are harmonic conjugates: level curves of $u$ and $v$ intersect at right angles — equipotentials and field lines form an orthogonal net.

A conformal map $w=f(z)$ (holomorphic with $f^{\prime }(z)\ne 0$) preserves Laplace's equation: if $V(x,y)$ is harmonic, then $V(x(u,v),y(u,v))$ is harmonic in $(u,v)$. This transforms a boundary-value problem on a complicated domain into one on a simpler domain without changing the governing equation.

Strategy. Given a simply connected domain $\mathcal{D}$ with boundary data, the Riemann mapping theorem guarantees a conformal map from $\mathcal{D}$ onto the unit disk. On the unit disk, the Dirichlet problem is solved by Poisson's integral formula:

$$V(r,\theta )=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{(1-r^2)V_0({\theta }^{\prime })}{1-2r\cos (\theta -{\theta }^{\prime })+r^2}d{\theta }^{\prime },$$

where $V_0(\theta )$ is the boundary data on the unit circle and $(r,\theta )$ are polar coordinates inside the disk. Composing with the conformal map solves the original problem.

Schwarz-Christoffel formula. For polygonal domains (boundaries consisting of straight-line segments), the Schwarz-Christoffel formula gives the conformal map from the upper half-plane to the polygon explicitly as an integral of a product of power functions. This is the workhorse tool for 2D electrostatics problems involving polygonal conductors (microstrip lines, coaxial cable cross-sections, etc.).

For domains that are not simply connected (annuli, exterior of obstacles), the mapping must be handled case-by-case. The group of conformal automorphisms of the unit disk is $\mathrm{PSU}(1,1)$ (Mobius transformations preserving the disk), and any two simply connected proper domains in $\mathbb{C}$ are conformally equivalent.

The Dirichlet principle [Master]

Variational characterisation

Among all functions $V$ on $\mathcal{V}$ that take prescribed Dirichlet boundary values, the one that satisfies Laplace's equation minimises the Dirichlet energy

$$W[V]:=\frac{{ϵ}_0}{2}{\int }_{\mathcal{V}}|\nabla V{|}^{2}d\tau .$$

To see this, let $V_0$ be the harmonic solution and let $\delta V$ be any smooth perturbation vanishing on $S$. Write $V=V_0+\delta V$. Then

$$W[V]=W[V_0]+{ϵ}_0{\int }_{\mathcal{V}}\nabla V_0\cdot \nabla (\delta V)d\tau +\frac{{ϵ}_0}{2}{\int }_{\mathcal{V}}|\nabla (\delta V){|}^{2}d\tau .$$

The cross term vanishes by Green's first identity together with ${\nabla }^2{V}_0=0$ and $\delta V{|}_S=0$:

$${\int }_{\mathcal{V}}\nabla V_0\cdot \nabla (\delta V)d\tau =-{\int }_{\mathcal{V}}\delta V{\nabla }^2{V}_{0}d\tau +{\oint }_S\delta V\frac{\partial V_0}{\partial n}da=0.$$

So $W[V]=W[V_0]+\frac{{ϵ}_0}{2}\int |\nabla (\delta V){|}^{2}d\tau \ge W[V_0]$, with equality only when $\delta V=0$ (since $\delta V{|}_S=0$ and $\nabla (\delta V)=0$ force $\delta V\equiv 0$). The harmonic function is the unique energy minimiser.

The converse is also true: if $V_0$ minimises $W$ among all functions with the given boundary data, then $V_0$ is harmonic. This follows by setting the first variation to zero. The Dirichlet principle establishes a one-to-one correspondence between solutions of Laplace's equation and minimisers of the electrostatic energy.

Historical note on the Dirichlet principle

Riemann (1851) used this minimisation principle freely, attributing it to Dirichlet's lectures, to prove existence of harmonic functions with given boundary data. Weierstrass (1870) objected: the argument assumes the infimum is attained by a smooth function, but there exist variational problems where the infimum is not achieved by any admissible function. His counterexample, a functional whose infimum is $-\infty$ but whose minimising sequence has no limit in the admissible class, showed that the "obvious" existence argument had a gap.

Hilbert (1900–1901) rescued the principle by developing the direct method of the calculus of variations: work in a suitable function space (what we now call the Sobolev space $H_0^1$), prove existence of a minimiser via weak compactness, show the minimiser satisfies the Euler-Lagrange equation (Laplace's equation) in a weak (distributional) sense, and finally invoke elliptic regularity theory to upgrade to a classical solution. This programme — variational existence, weak solutions, regularity — became the template for 20th-century PDE theory. The Sobolev and Banach-space machinery it motivated is now standard in every graduate analysis course.

Capacity

The electrostatic capacity (or capacitance) of a conductor $\mathcal{C}$ is

$$C=\frac{Q}{V}=\frac{{ϵ}_0}{\underset{\begin{array}{c}u{|}_{\partial \mathcal{C}}=1\\ u\to 0\text{ at }\infty \end{array}}{\inf }{\int }_{{\mathbb{R}}^3\setminus \mathcal{C}}|\nabla u{|}^{2}d\tau },$$

where the infimum is taken over all functions that equal 1 on $\partial \mathcal{C}$ and vanish at infinity. By the Dirichlet principle, the infimum is attained by the equilibrium potential — the harmonic function satisfying these boundary conditions. For a sphere of radius $R$, direct calculation gives $C=4\pi {ϵ}_{0}R$.

Capacity measures how much charge a body can hold at unit potential. It is a conformal invariant in 2D (by the invariance of the Dirichlet energy under conformal maps, Exercise 10), and connects electrostatics to potential theory, probability, and geometric measure theory. The logarithmic capacity in 2D plays the role of the Newtonian capacity in 3D.

Harmonic measure

Let $\mathcal{D}$ be a bounded domain with boundary $S=S_1\cup S_2\cup \cdots \cup S_n$ (disjoint pieces). The harmonic measure ${\omega }_i(\mathbf{r})$ is the harmonic function in $\mathcal{D}$ equal to 1 on $S_i$ and 0 on the remaining pieces. By the maximum principle, $0\le {\omega }_i\le 1$ and ${\sum }_i{\omega }_i=1$ — the harmonic measures form a probability distribution over boundary components at each interior point.

Probabilistic interpretation (Kakutani, 1944): ${\omega }_i(\mathbf{r})$ equals the probability that a Brownian motion started at $\mathbf{r}$ first exits $\mathcal{D}$ through $S_i$. The Dirichlet problem is equivalent to computing exit probabilities for Brownian motion. This bridges electrostatics, potential theory, and stochastic analysis: the Laplacian is both the operator governing electrostatic potentials and the infinitesimal generator of Brownian motion. The connection runs deep — virtually every classical result about harmonic functions has a probabilistic counterpart, and vice versa.

Historical notes [Master]

Pierre-Simon Laplace introduced the equation bearing his name in Théorie des attractions des sphéroïdes et de la figure des planètes (1782, published 1785), in the context of gravitational potentials of celestial bodies. He wrote the operator in spherical coordinates and used it to study the external gravitational field. The Cartesian form ${\partial }^{2}V/\partial x^2+{\partial }^{2}V/\partial y^2+{\partial }^{2}V/\partial z^2=0$ was written out explicitly by Legendre and Laplace in the 1780s.

George Green's An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (1828) introduced Green's functions and Green's identities, providing the general framework for boundary-value problems. Green, a self-taught miller's son from Nottingham, published the essay at his own expense; it was largely ignored until Lord Kelvin (William Thomson) rediscovered it in 1845 and recognised its power.

The method of images was developed by William Thomson in 1845 for the theory of telegraphy, specifically to compute the capacitance of telegraph cables. His insight — that a point charge near a conducting sphere can be replaced by an image charge — turned an intractable boundary-value problem into elementary algebra.

The Dirichlet principle was used by Riemann in his 1851 dissertation and his 1857 papers on Abelian functions to prove existence of harmonic functions with prescribed boundary data. Weierstrass's critique (published 1870, circulated earlier) showed the variational argument had a gap. Hilbert's 1901 paper on the Dirichlet principle inaugurated the direct method of the calculus of variations and led to the development of Sobolev spaces, weak solutions, and modern elliptic PDE theory. Poisson's integral formula for the disk (1820) and Thomson's method of images (1845) are the two constructive pillars that Green's framework unifies.

Connections to other fields [Master]

Quantum mechanics

The time-independent Schrödinger equation $-\frac{{\hslash }^2}{2m}{\nabla }^2\psi +V\psi =E\psi$ reduces to Laplace's equation in the free-particle, zero-energy limit ($V=0$, $E=0$). The eigenfunction expansion of the Dirichlet Green's function developed above is identical in structure to the resolution of the identity in quantum mechanics: $G_D(\mathbf{r},{\mathbf{r}}^{\prime })={\sum }_n{\psi }_n(\mathbf{r}){\psi }_n({\mathbf{r}}^{\prime })/{\lambda }_n$ mirrors the spectral representation of the resolvent $(H-E{)}^{-1}$. The separation-of-variables technique for the Laplacian in spherical coordinates produces the same angular eigenfunctions (Legendre polynomials, spherical harmonics) that appear as angular-momentum eigenstates in the hydrogen atom.

Complex analysis

In 2D, harmonic functions are real parts of holomorphic functions. This equivalence makes conformal mapping a complete solution theory for 2D electrostatics: the Riemann mapping theorem guarantees that any simply connected domain can be mapped to the disk, where Poisson's formula solves the Dirichlet problem. The Cauchy-Riemann equations simultaneously encode Laplace's equation and the orthogonality of equipotentials and field lines. The theory of analytic continuation, harmonic conjugates, and boundary behaviour of holomorphic functions all have direct electrostatic readings.

Probability and stochastic analysis

The Laplacian is the infinitesimal generator of Brownian motion. The harmonic measure ${\omega }_i(\mathbf{r})$ equals the probability that a Brownian path started at $\mathbf{r}$ first exits the domain through $S_i$ (Kakutani's theorem, 1944). The Dirichlet problem $-$ finding a harmonic function with given boundary data $-$ is equivalent to computing the expected value of the boundary data at the Brownian exit point. This connection makes Monte Carlo methods (walking random walks to estimate exit probabilities) a viable numerical technique for boundary-value problems, and it underlies the modern probabilistic approach to potential theory.

General relativity and differential geometry

The flat-space Laplacian ${\nabla }^2$ generalises to the Laplace-Beltrami operator ${\Delta }_g=\frac{1}{\sqrt{|g|}}{\partial }_i(\sqrt{|g|}{g}^{ij}{\partial }_j)$ on a Riemannian manifold $(M,g)$. The uniqueness theorems, maximum principle, and Dirichlet principle all survive this generalisation; the proofs carry over with the volume element $\sqrt{|g|}{d}^{n}x$ replacing the Euclidean measure. In general relativity 13.02.01, harmonic coordinates (coordinates satisfying ${\Delta }_g{x}^{\mu }=0$) are used to fix gauge freedom in the Einstein equations, and scalar fields on curved spacetimes satisfy the curved-space Klein-Gordon equation, whose spatial part is the Laplace-Beltrami operator.

Computational methods

The Dirichlet principle — minimise $W[V]=\frac{{ϵ}_0}{2}\int |\nabla V{|}^{2}d\tau$ — is the theoretical foundation of the finite element method (FEM) for electrostatics. The domain is discretised into a mesh of elements, the potential is approximated by piecewise polynomial basis functions, and the variational problem reduces to a sparse linear system $K\mathbf{u}=\mathbf{f}$ where $K$ is the stiffness matrix (discrete analogue of $-{\nabla }^2$). The convergence of FEM solutions to the true harmonic function is guaranteed by the same elliptic regularity theory that rescued the Dirichlet principle from Weierstrass's objection.

Bibliography [Master]

• Laplace, P.-S. (1785). "Théorie des attractions des sphéroïdes et de la figure des planètes." Mémoires de l'Académie royale des Sciences de Paris, 1782 (published 1785), 113–196. Origin of Laplace's equation in the gravitational context.

• Green, G. (1828). An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. Nottingham: privately printed. Introduction of Green's functions and Green's identities. Reprinted in Mathematical Papers of George Green, ed. N. M. Ferrers (1871).

• Thomson, W. (Lord Kelvin) (1845). "Extraits de deux mémoires sur le développement des fonctions de Laplace et de Lamé." Journal de mathématiques pures et appliquées, 10, 213–224. Rediscovery of Green's work; development of the method of images.

• Riemann, B. (1851). "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse." Inaugural dissertation, Göttingen. Use of the Dirichlet principle in complex analysis.

• Weierstrass, K. (1870). "Über das sogenannte Dirichlet'sche Princip." Mathematische Werke, II, 49–54. Critique of the existence argument in the Dirichlet principle.

• Hilbert, D. (1901). "Über das Dirichlet'sche Princip." Jahresbericht der Deutschen Mathematiker-Vereinigung, 8, 184–188. Rehabilitation of the Dirichlet principle via the direct method.

• Kakutani, S. (1944). "Two-dimensional Brownian motion and harmonic functions." Proceedings of the Imperial Academy, Tokyo, 20, 706–714. Probabilistic representation of harmonic measure.

• Griffiths, D. J. (2017). Introduction to Electrodynamics, 4th ed. Cambridge University Press. Ch. 3 (Potentials): accessible treatment of Laplace's equation, separation of variables, and images.

• Jackson, J. D. (1999). Classical Electrodynamics, 3rd ed. Wiley. Ch. 2 (Boundary-Value Problems in Electrostatics I): Green's functions, eigenfunction expansions, and image methods at graduate level.

• Zangwill, A. (2013). Modern Electrodynamics. Cambridge University Press. Ch. 4: comprehensive treatment of potential theory including conformal mapping and variational methods.