Uniqueness Theorems for the Potential

Every method in this chapter ends the same way: a function is produced — by mirror trickery, by infinite series, by a pile of relaxed grid numbers — and declared to be the potential. Declared on what authority? A differential equation has infinitely many solutions; matching a boundary condition narrows the field, but who says it narrows it to one? The uniqueness theorems do, and they are the legal foundation on which the entire boundary-value edifice stands. Without them, the method of images would be a heuristic; with them, it is a proof. The argument is one of the most elegant in physics. Suppose two rival potentials both solve the same Poisson equation with the same boundary data. Subtract them: the sources cancel, the boundary data cancel, and what remains is a harmonic function $W$ with nothing — zero on the boundary, zero charge inside. Green's first identity then performs the execution: it equates the volume integral of $|\nabla W|^2$ , a manifestly non-negative quantity (it is, up to $\varepsilon_0/2$ , the field energy of the discrepancy), to a boundary term built from exactly the data that vanished. A non-negative integral equal to zero has a vanishing integrand: $\nabla W = 0$ everywhere, the rivals were the same solution wearing different formulas, and the case is closed. The chapter needs two versions. The first theorem covers prescribed boundary potentials — electrodes, grounded planes, infinity. The second covers the lab's awkward customer: isolated conductors whose total charge is known but whose potential and surface distribution are not. Both proofs fit on one page, and both are constantly, silently at work — including in the answer to a question you may never have thought to ask: how do you know the charge on an isolated sphere spreads uniformly?

Key takeaways

First uniqueness theorem: the solution of $\nabla^2V=-\rho/\varepsilon_0$ in a region is unique once $V$ is specified on the entire boundary (infinity counts, with $V\to0$ ). Neumann data do the same up to an additive constant.
Second uniqueness theorem: with conductors present, specifying the total charge on each conductor (plus $\rho$ between them) determines the field uniquely — no knowledge of the surface distributions required.
One proof engine: the difference $W=V_1-V_2$ of two candidates is harmonic with null data; Green's first identity forces $\int|\nabla W|^2\,\mathrm{d}\tau = 0$ , hence $\nabla W=0$ .
The integral $\tfrac{\varepsilon_0}{2}\int|\nabla W|^2$ is the field energy of the discrepancy — uniqueness says a difference field with no sources and no boundary presence cannot afford to exist.
Practical payoff: the license to guess — any function that is checked against the equation and the boundary data is not an answer but the answer; and any two correct methods must produce identical functions, however different their formulas look.
KEY IDEA: same equation + same boundary data ⇒ same field; the difference of rivals is a source-free, boundary-free field whose energy integral must vanish — so images, series, grids, and lucky guesses all converge on one truth.

The core insight

Two solutions of the same boundary-value problem differ by a harmonic function with null boundary data, and Green's first identity forces that difference's field energy — $\int|\nabla W|^2$ — to zero: the solution is unique (up to a constant for pure Neumann data), which is the license behind every guessing method in this chapter.

Where this lives today

Uniqueness is modern computational physics' quality-assurance department. The standard validation ritual for a new field solver — "method of manufactured solutions" — plants an analytic potential, derives the source and boundary data it implies, and demands the code reproduce it; the demand is only meaningful because the reproduced answer has no alternative. Cross-validation between methods (finite elements vs. boundary elements vs. the walk-on-spheres Monte Carlo of the mean-value article) leans on the same guarantee: disagreement beyond discretization error is proof of a bug, not a difference of opinion. And EMC engineers certifying a shielded enclosure are cashing the one-line corollary: boundary at one potential, charge-free interior, hence field-free — no simulation required. The theorems' deepest modern echo is in inverse problems. Impedance tomography, geophysical prospecting, and non-destructive testing all ask uniqueness questions one derivative harder — does boundary data determine an interior coefficient? — and the affirmative answers (Calderón problem, 1980s–2000s) are direct descendants of the Green-identity arguments on this page. Even machine-learned PDE surrogates get audited against uniqueness: a network that returns two different fields for one boundary condition has learned physics that provably does not exist.

Origins · 1 minBernhard Riemann and how this was discovered

Bernhard Riemann

1826–1866 · Göttingen

Riemann was Dirichlet's greatest student and the man who bet the largest mathematical structures of his century on the integral at the heart of this article. In his 1851 thesis he needed harmonic functions to exist on arbitrary domains, and argued they must: among all candidates with the given boundary values, something should minimize the energy $\int|\nabla W|^2$ — and the minimizer is harmonic. He named the argument the Dirichlet principle, after his teacher's lectures, and built complex analysis and the theory of Riemann surfaces on top of it. Then in 1870 Weierstrass exhibited variational problems whose infimum no function attains, and the principle — with everything resting on it — stood discredited for thirty years, its conclusions believed but unproven. Hilbert's 1900–1904 rescue, supplying the compactness the argument had silently assumed, turned the principle into a theorem and inaugurated the modern calculus of variations and the function spaces every finite-element solver now lives in. The energy integral that convicts our difference function is the same object: uniqueness is the easy, always-true half of the Dirichlet principle — the half even Weierstrass couldn't dent.

History

1828

— Green's Essay proves boundary values determine the interior potential — the first uniqueness theorem in recognizable form

1840

— Gauss connects existence and uniqueness to energy minimization for inverse-square potentials

1851

— Riemann's thesis christens the Dirichlet principle and stakes complex analysis on the energy integral

1870

— Weierstrass shows minimizers need not exist, breaking the existence half while uniqueness stands untouched

1904

— Hilbert rehabilitates the Dirichlet principle rigorously, founding the variational framework behind modern solvers

The physical picture

Run the courtroom. Two theorists submit potentials $V_1$ and $V_2$ for the same problem — same charges in the region, same values on every boundary surface. The judge doesn't examine either candidate; she examines their disagreement, $W = V_1 - V_2$ . Whatever the rivals share, the subtraction destroys: the charge density cancels ( $W$ is harmonic), the boundary values cancel ( $W = 0$ on every wall). The disagreement is a field over nothing, anchored to nothing. The theorems say such a field cannot be nonzero anywhere — disagreement between correct answers is impossible, so there is exactly one correct answer.

Figure

The difference-function argument — interrogate the disagreement, not the candidates

The best physical reading of the verdict is energetic. The quantity $\tfrac{\varepsilon_0}{2}\int|\nabla W|^2\,\mathrm{d}\tau$ is the field energy the disagreement would carry (the energy-density articles of E1 built that interpretation). Green's identity says this energy equals a surface term assembled from $W$ and its slope on the boundary — the exact quantities the subtraction annihilated. A discrepancy field would be energy with no source to have paid for it and no boundary to have imported it. Electrostatics does not extend that kind of credit. Notice what the argument does not need: any formula for either candidate, any symmetry, any coordinate system. That is why the theorem certifies solutions found by methods that haven't been invented yet — it audits results, not procedures. And notice its practical converse, sharpened in the warning below: when two legitimate methods hand you wildly different-looking expressions, uniqueness doesn't ask you to pick; it asserts they are the same function and turns the comparison into a free consistency check. You have met a preview: the E1 Laplace article proved the charge-free Dirichlet case from the maximum principle ("boundary data determine the interior"). This article supplies the industrial-strength versions — with charge present, with Neumann data, with the boundary at infinity handled honestly, and with the second theorem's conductors, where even the boundary potentials are unknown and only total charges are given.

The theorems and the tool

The statements, in the order the chapter uses them:

\textbf{First:}\quad \nabla^2V = -\rho/\varepsilon_0 \text{ in } \Omega, \quad V \text{ given on all of } \partial\Omega \;\;\Longrightarrow\;\; V \text{ unique} \qquad (1)

where: $\partial\Omega$ — every bounding surface, the sphere at infinity included (with $V\to0$ there for localized sources). The Neumann variant — $\partial V/\partial n$ given instead — yields uniqueness up to an additive constant, the floating reference of the previous article; mixed data (some walls Dirichlet, some Neumann) give full uniqueness so long as at least one piece is Dirichlet.

\textbf{Second:}\quad \rho \text{ given between conductors, total charge } Q_i \text{ given on each} \;\;\Longrightarrow\;\; \mathbf{E} \text{ unique} \qquad (2)

where: $Q_i$ — the net charge on conductor $i$ , with no information about its distribution or its potential. This is the theorem for the isolated, floating conductor — the customer the Dirichlet/Neumann taxonomy couldn't seat. The tool for both is Green's first identity, which is integration by parts in three dimensions — take the product rule $\nabla\cdot(W\nabla W) = |\nabla W|^2 + W\nabla^2W$ and integrate over $\Omega$ with the divergence theorem:

\int_\Omega |\nabla W|^2\,\mathrm{d}\tau \;=\; \oint_{\partial\Omega} W\,\frac{\partial W}{\partial n}\,\mathrm{d}a \;-\; \int_\Omega W\,\nabla^2W\,\mathrm{d}\tau \qquad (3)

where: $W$ — any sufficiently smooth function; $\partial W/\partial n$ — its outward normal derivative. Read the three pieces physically for the difference function to come: the left side is (twice, over $\varepsilon_0$ ) the energy of the discrepancy field; the first right-hand term is what the boundary knows about the discrepancy; the last term is what the sources know. The proofs consist of arranging for the right side to die.

Derivation AFirst uniqueness theorem — Dirichlet, Neumann, and the boundary at infinity

We prove statement (1) and its Neumann sibling in one pass, then close the loophole that most presentations wave at: the "boundary" of an unbounded region. The strategy is the difference-function ritual — subtract, observe the cancellations, deploy identity (3), and squeeze. Watch which hypothesis kills which term; that bookkeeping is exactly what distinguishes the Dirichlet, Neumann, and infinity cases.

Step 1. Let $V_1, V_2$ both satisfy $\nabla^2V_i = -\rho/\varepsilon_0$ in $\Omega$ with identical boundary data. Set $W = V_1 - V_2$ . Subtracting the equations: $\nabla^2W = 0$ — the same $\rho$ cancels, which is why the theorem covers Poisson, not just Laplace.

Step 2. Apply identity (3) to $W$ ; the source term dies immediately ( $\nabla^2W=0$ ):

\int_\Omega |\nabla W|^2\,\mathrm{d}\tau = \oint_{\partial\Omega} W\,\frac{\partial W}{\partial n}\,\mathrm{d}a.

Step 3. Kill the boundary term with the data. Dirichlet: both candidates equal $g$ on $\partial\Omega$ , so $W=0$ there — the integrand vanishes through its first factor. Neumann: both have the same normal slopes, so $\partial W/\partial n = 0$ — it vanishes through its second. Either way,

\int_\Omega |\nabla W|^2\,\mathrm{d}\tau = 0.

Step 4. Squeeze: the integrand is non-negative, so it must vanish pointwise — $\nabla W = 0$ , $W$ constant throughout $\Omega$ . Dirichlet: the constant is its boundary value, zero, so $V_1 = V_2$ identically. Neumann: no condition pins the constant — $V_1 = V_2 + c$ , uniqueness up to the floating reference, exactly as the data deserve. Why the argument is airtight and not circular: nothing was assumed about how $V_1, V_2$ were obtained, and the only analytic fact used is that a non-negative continuous integrand with zero integral vanishes.

Step 5. The infinity loophole. For an unbounded region, apply the argument in a ball of radius $R$ and send $R\to\infty$ ; the extra boundary piece is the giant sphere, contributing $\oint W\,\partial_r W\,\mathrm{d}a$ . For localized sources with $V\to0$ at infinity, each candidate falls at least as fast as $1/R$ (leading multipole), so $W \sim 1/R$ , $\partial_rW \sim 1/R^2$ , while the sphere's area grows as $R^2$ : the term scales as $R^2\cdot\tfrac1R\cdot\tfrac{1}{R^2} = 1/R \to 0$ . The vacuum condition " $V\to0$ " is thus not decoration — it is the boundary datum that closes the proof for open space.

Figure

The second theorem's arena — conductors that only tell you their totals

Go deeper · Derivation B · 2 minSecond uniqueness theorem — conductors with specified total charges

Derivation BSecond uniqueness theorem — conductors with specified total charges

Now the lab's real configuration: a set of isolated conductors, conductor $i$ carrying known net charge $Q_i$ , with prescribed charge density $\rho$ in the space between, and (say) vacuum out to infinity. No conductor's potential is known; no surface distribution is known. We prove the field is nonetheless unique. The new difficulty is that the difference function no longer vanishes on the conductor surfaces — it is merely constant on each. The rescue is that constants pull out of integrals, and the total-charge data kill what remains. This is the theorem that lets you assert, rather than assume, how charge arranges itself on isolated metal.

Step 1. Let two field configurations $\mathbf{E}_1, \mathbf{E}_2$ both satisfy: Gauss's law with the given $\rho$ between the conductors, equipotential conductor surfaces, and $\oint_{S_i}\mathbf{E}\cdot\mathrm{d}\mathbf{a} = Q_i/\varepsilon_0$ over each conductor $i$ . Write $W = V_1 - V_2$ for the corresponding potentials: as before $\nabla^2W = 0$ in the region between conductors.

Step 2. On each conductor surface $S_i$ , both $V_1$ and $V_2$ are constant (equipotentials), so $W$ equals some constant $W_i$ there — unknown, and not necessarily zero. This is where the first theorem's argument stalls.

Step 3. Deploy identity (3) over the region between the conductors (the outer boundary at infinity dies as in Derivation A). The boundary is now the union of conductor surfaces:

\int |\nabla W|^2\,\mathrm{d}\tau = \sum_i \oint_{S_i} W\,\frac{\partial W}{\partial n}\,\mathrm{d}a = \sum_i W_i \oint_{S_i} \frac{\partial W}{\partial n}\,\mathrm{d}a,

the constants $W_i$ stepping outside their integrals — the move the equipotential property exists to enable.

Step 4. Evaluate each flux integral with the total-charge data. Since $\partial W/\partial n = \partial V_1/\partial n - \partial V_2/\partial n$ and each candidate's normal derivative integrates over $S_i$ to $\mp Q_i/\varepsilon_0$ (same $Q_i$ for both — that is the hypothesis),

\oint_{S_i} \frac{\partial W}{\partial n}\,\mathrm{d}a = 0 \quad\text{for every } i.

Why totals suffice where distributions weren't needed: the identity only ever asks each surface for the product (constant) × (total flux). The pointwise distribution never enters — which is precisely why the theorem can certify solutions when $\sigma_i(\mathbf{r})$ is part of the unknown.

Step 5. The right side collapses to zero; squeeze as before: $\nabla W = 0$ , so $\mathbf{E}_1 = \mathbf{E}_2$ everywhere — the field, the surface densities ( $\sigma = \varepsilon_0E_\perp$ ), and the conductor potentials up to the single overall constant (pinned by $V\to0$ at infinity if present). Feed each conductor its coulombs and nature has no further choices to make.

Limiting cases

Data supplied	Uniqueness delivered	Notes
$V$ on all boundaries (Dirichlet)	full	the chapter's default license
$\partial V/\partial n$ on all boundaries (Neumann)	up to additive constant	budget constraint must hold first
Mixed (some walls $V$ , some slope)	full (if any Dirichlet piece)	worked sandwich of the BC article
Conductors with total charges $Q_i$	field fully unique	second theorem; distributions are outputs
Unbounded region, $V\to0$ at infinity	full	surface term dies like $1/R$
Charge-free region, boundary all one value	$V$ = that constant everywhere	the Faraday-cage one-liner

Common trapTwo methods gave different formulas, so at least one must be wrong

Uniqueness says the opposite: if both candidates genuinely satisfy the same equation and the same boundary data, they are the same function — even when one is a two-term image expression and the other an infinite Legendre series with no visible resemblance. The theorems certify functions, not notations. This converts superficial disagreement into a free, brutal consistency test: evaluate both expressions at a few points; any true mismatch means one construction failed a hypothesis (a missed boundary piece, a source in the wrong region, a dropped infinity condition) — that's where to hunt, not in the algebra of the other method. Professionals validate solvers exactly this way: analytic images versus numerical grids, forced to agree to machine precision by a theorem.

Worked examples

1 · The license in action — certifying a one-line guess (confidence builder)

A conducting sphere of radius $R = 5.0\,\mathrm{cm}$ is held at $V_0 = 1000\,\mathrm{V}$ in empty space. Guess, certify, and harvest.

Step 1. Guess $V(r) = V_0R/r$ outside the sphere. Certification checklist, per the first theorem: (i) harmonic in the region? $V \propto 1/r$ is the radial Laplace solution — yes, for $r>R$ , where there is no charge. (ii) Boundary data? At $r=R$ : $V = V_0$ ✓; at infinity: $V \to 0$ ✓. All hypotheses met — this is the solution, and no further justification is owed or possible.

Step 2. Harvest the field and charge:

E(R) = -\frac{\partial V}{\partial r}\bigg|_R = \frac{V_0}{R} = \frac{1000}{0.050} = 2.0\times10^{4}\,\mathrm{V/m}, \qquad \sigma = \varepsilon_0E = 1.77\times10^{-7}\,\mathrm{C/m^2},

Q = 4\pi R^2\sigma = 4\pi\varepsilon_0RV_0 = 4\pi(8.854\times10^{-12})(0.050)(1000) = 5.6\times10^{-9}\,\mathrm{C}.

Step 3. Note what the theorem quietly excluded: solutions with hidden $r^\ell$ growing terms (killed by the infinity datum), lopsided angular pieces (they would violate the constant boundary value), and every other rival. One guess, one checklist, $Q = 5.6\,\mathrm{nC}$ — and the empty-cavity theorem is the same logic at speed: wall all at one value $\Rightarrow$ constant matches $\Rightarrow$ field zero.

2 · Why is the charge on an isolated sphere uniform? (exam level)

An isolated conducting sphere of radius $R = 2.0\,\mathrm{cm}$ , far from everything, carries total charge $Q = +2.0\,\mathrm{nC}$ — total specified, distribution not. Prove the distribution is uniform, then compute the surface density and potential, and state exactly which theorem did the work.

Step 1. Frame it as the second theorem's hypothesis: one conductor, given total $Q$ , $\rho=0$ outside, $V\to0$ at infinity. The theorem asserts: exactly one field configuration is consistent with these data.

Step 2. Exhibit one: the spherically symmetric candidate — uniform $\sigma = Q/4\pi R^2$ , exterior field $Q/4\pi\varepsilon_0r^2$ , constant potential on the metal. Check its credentials: harmonic outside ✓, sphere equipotential ✓, total flux $Q/\varepsilon_0$ ✓, dies at infinity ✓.

Step 3. Invoke uniqueness: since a valid configuration exists and only one can, the symmetric one is the one. Any imagined rival — charge clumped toward one pole, say — is not merely unstable or unlikely; it is mathematically excluded, because it would be a second solution to a problem the theorem allows only one. (Note the logic's direction: symmetry made the candidate easy to find, but uniqueness is what forbids asymmetric rivals — without it, "by symmetry" is an aesthetic, not an argument.)

Step 4. Harvest the numbers:

\sigma = \frac{Q}{4\pi R^2} = \frac{2.0\times10^{-9}}{4\pi(0.020)^2} = 4.0\times10^{-7}\,\mathrm{C/m^2}, \qquad V_{\text{sphere}} = \frac{Q}{4\pi\varepsilon_0R} = \frac{(8.99\times10^{9})(2.0\times10^{-9})}{0.020} = 9.0\times10^{2}\,\mathrm{V}.

The sphere sits at $899\,\mathrm{V}$ with $0.40\,\mu\mathrm{C/m^2}$ everywhere — "everywhere" being a theorem, not an assumption. Bring another charged body nearby and the second theorem still guarantees a unique (now non-uniform) answer — which is exactly the problem the image-charge articles will solve in closed form.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §3.1.5–3.1.6 · Jackson, Classical Electrodynamics, 3rd ed., §1.8–1.9 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §3.3 · Zangwill, Modern Electrodynamics, §7.3. Green's first identity used throughout is $\int_{\mathcal{V}}(\phi\nabla^2\psi+\nabla\phi\cdot\nabla\psi)\mathrm{d}\tau=\oint_{\mathcal{S}}\phi\nabla\psi\cdot\mathrm{d}\mathbf{a}$ , applied here with $\phi=\psi=W$ . The far-field scaling underpinning the open-space case is $W\sim1/r$ , $\nabla W\sim1/r^2$ , so the surface term scales as $\sim1/R\to0$ . The value $\varepsilon_0=8.854\times10^{-12}\ \mathrm{F/m}$ is the CODATA standard. Key worked numbers: $Q=4\pi\varepsilon_0RV_0=5.6\,\mathrm{nC}$ ; $\sigma=Q/4\pi R^2=4.0\times10^{-7}\,\mathrm{C/m^2}$ ; $V=kQ/R=899\,\mathrm{V}$ .

Builds on: Boundary Conditions for the Potential · Harmonic Functions and the Mean-Value Theorem · Laplace's Equation (E2) · Poisson's Equation (E2)

Leads to: Method of Images · Image Charge for a Grounded Plane · Image Charges for a Conducting Sphere · Separation of Variables in Cartesian Coordinates

Guessing a solution still feels like cheating. Where exactly does the rigor come in?

In the checklist, which is the entire proof. "Guess" describes only how the candidate was found — psychology, not mathematics. Rigor enters when you verify the three hypotheses: the candidate satisfies the correct Poisson/Laplace equation in the region, matches every piece of boundary data (infinity included), and the problem type is covered by a uniqueness theorem. Those checks are exactly what the theorem needs; once they pass, "there is no other solution" is a proved statement. The method of images is respectable for precisely this reason — its mirror charges are a search strategy, and uniqueness converts the found object into the answer.

How is this related to the E1 Laplace article's proof that boundary data determine the interior?

Same destination, different vehicle, smaller cargo. E1 used the mean-value/maximum-principle route: a harmonic difference with zero boundary values can have no interior extremum, hence vanishes — clean, but naturally suited to the charge-free Dirichlet case. The Green-identity route here handles Poisson sources, Neumann data, unbounded regions, and — decisively — the second theorem's conductors, where the difference is only constant on each surface and one needs the constants-out-of-integrals maneuver. Two proofs of the overlap case is also insurance you'll use: they fail differently when hypotheses are violated, which helps diagnose which hypothesis a broken problem lost.

Why does the Neumann case only give uniqueness up to a constant, and does it ever matter?

Because no Neumann datum ever mentions the potential's value — only slopes — and adding a constant changes no slope; the proof honestly reports this by delivering $\nabla W=0$ and then finding no condition to evaluate the constant. It "matters" only bureaucratically: fields, charges, forces, and capacitances are all constant-blind. It bites in practice exactly once per simulation — an all-Neumann linear system is singular until you pin one node — and in bookkeeping, when two references must be reconciled before potentials are compared.

Do the theorems survive dielectrics?

Yes, with the identity upgraded. For linear media the equation is $\nabla\cdot(\varepsilon\nabla V) = -\rho_f$ , and the proof runs on the weighted integral $\int \varepsilon\,|\nabla W|^2\,\mathrm{d}\tau$ , using $\nabla\cdot(W\varepsilon\nabla W)$ ; positivity survives because $\varepsilon>0$ , and the interface matching conditions ( $V$ and $\varepsilon\,\partial V/\partial n$ continuous across uncharged dielectric boundaries) make the internal surface terms cancel in pairs. The E3 chapter uses this silently every time it "guesses" the field of a dielectric sphere. Nonlinear media are another world: multiple stable solutions genuinely occur (ferroelectric hysteresis is physical non-uniqueness), and the linear theorems correctly decline to apply.

Can I use the first theorem when I know the potential on some conductors and only the total charge on others?

Yes — the two theorems compose. The Green-identity machinery processes each boundary piece independently: Dirichlet pieces kill their surface term via $W=0$ , specified-total-charge conductors kill theirs via the constant-times-zero-flux step of Derivation B. Any mixture of {prescribed $V$ } and {prescribed total $Q$ } conductors, plus given $\rho$ in between, yields a unique field. This mixed form is the everyday one: a lab setup with one driven electrode, several grounded surfaces, and a floating guard ring is exactly such a hybrid.

Does uniqueness say anything about stability — that small changes in boundary data cause small changes in the answer?

Not by itself; that is the third leg of Hadamard's well-posedness (existence, uniqueness, continuous dependence), introduced in the boundary-conditions article. For the legal data types the same energy machinery does deliver stability — the difference of solutions for slightly different Dirichlet data has energy controlled by the data difference. For illegal pairings (Cauchy data) uniqueness can technically hold while stability fails catastrophically, which is the deep reason the taxonomy forbids them: an answer that exists, is unique, and is wrong by megavolts when the data wobble by microvolts is not an answer.

Is there a variational reading of uniqueness?

A tight one — it is the trivial half of the Dirichlet principle. Among all functions with the prescribed boundary values, the true potential minimizes the field energy $\tfrac{\varepsilon_0}{2}\int|\nabla V|^2$ (Thomson's theorem is the conductor version). Strict convexity of that energy in $\nabla V$ means a minimizer, if it exists, is unique — precisely our theorem in variational clothing. The historically treacherous half was existence of the minimizer (Weierstrass's objection, Hilbert's repair); uniqueness never wavered. Finite-element solvers literally minimize this energy over trial functions, so every FEM run is a small homage to the principle.