Boundary Conditions for the Potential

Every solver in this chapter — guessing, images, series, grids — begins the same way: "given the boundary conditions…". This article interrogates that innocent phrase. What, exactly, are you allowed to prescribe on the rim of a region so that Laplace's or Poisson's equation has a solution at all, has only one, and doesn't shatter when the data wiggle? The question sounds bureaucratic and is anything but: get the data menu wrong and you either starve the equation (infinitely many solutions), overfeed it (no solution), or poison it (a "solution" so unstable that rounding error in the data rewrites the answer in the interior). The menu turns out to be short and strict. You may prescribe the potential itself on the boundary — Dirichlet data, what a wired-up electrode provides. Or you may prescribe its normal derivative — Neumann data, which by $\sigma = -\varepsilon_0\,\partial V/\partial n$ is really a statement about surface charge, with a global catch: the prescribed flux must balance the enclosed charge, or no solution exists, and even then the answer floats by an additive constant. You may mix the two on different pieces of the boundary. What you may not do is prescribe both on the same piece: Cauchy data, the natural diet of initial-value problems, overdetermines an elliptic equation — and worse, as Hadamard's famous example shows with two lines of algebra, the Cauchy problem for Laplace's equation amplifies invisible boundary noise into exponential interior garbage. The E1 article on boundary conditions derived how fields cross charged interfaces; this one legislates what may be prescribed on boundaries. Together they are the rulebook under which every construction in the rest of the chapter — starting with the uniqueness theorems next door — legally operates.

Key takeaways

A second-order elliptic problem needs exactly one condition at each boundary point: the potential (Dirichlet) or its normal derivative (Neumann) — or Dirichlet on some pieces and Neumann on others (mixed).
Physics supplies the data: wired electrodes fix $V$ (Dirichlet); known surface charge or insulation fixes $\partial V/\partial n$ (Neumann, since $\sigma=-\varepsilon_0\,\partial V/\partial n$ ); symmetry planes give the free Neumann condition $\partial V/\partial n = 0$ .
Pure Neumann data carry a compatibility constraint — total prescribed flux must equal $-Q_{\text{enc}}/\varepsilon_0$ (Gauss's law in potential clothing) — and determine $V$ only up to an additive constant.
Prescribing $V$ and $\partial V/\partial n$ together (Cauchy data) on the same surface overdetermines the problem, and the Laplace–Cauchy problem is catastrophically unstable: boundary ripples of size $1/n$ grow like $e^{nL}$ inside (Hadamard).
For localized systems, infinity is part of the boundary: $V\to0$ there is the condition that kills growing solutions and closes the data budget.
KEY IDEA: one condition per boundary point — value or slope, never both; Neumann data must balance the charge budget and leave the zero of potential floating; anything more than the budget destroys existence or stability.

The core insight

Well-posed electrostatics prescribes exactly one datum — $V$ (Dirichlet) or $\partial V/\partial n$ (Neumann) — on each piece of the boundary, with Neumann data obeying the Gauss compatibility budget and fixing $V$ only up to a constant; Cauchy data (both at once) overdetermine and destabilize the problem.

Where this lives today

Open any commercial field solver — COMSOL, Ansys, or the open-source FEniCS — and the boundary-condition dialog is this article as a dropdown menu: Dirichlet ("electric potential"), Neumann ("surface charge density" / "zero charge"), Robin ("distributed impedance"), plus the symmetry-plane shortcut that computes half a device and mirrors it through a free Neumann wall. The floating-conductor hybrid has its own checkbox ("floating potential"), and all-Neumann models still make beginners rediscover the floating constant when their solver reports a singular matrix until one node is grounded. The deepest modern twist inverts the whole subject. Instead of prescribing one datum per point, measure both on an accessible boundary and ask what interior would link them: the Dirichlet-to-Neumann map. Electrical impedance tomography drives currents through electrodes on a torso (Neumann), measures the resulting voltages (Dirichlet), and reconstructs the conductivity map of the tissue inside — Calderón's inverse problem, whose mathematics (when is the map invertible? how unstably?) descends directly from Hadamard's observation that inward continuation is exponentially ill-posed. The instability never went away; imaging science learned to regularize it and made it a diagnostic tool.

Origins · 1 minPeter Gustav Lejeune Dirichlet and how this was discovered

Peter Gustav Lejeune Dirichlet

1805–1859 · Berlin and Göttingen

Dirichlet was the mathematician other mathematicians went to when they wanted to know what a proof was. At twenty he settled a case of Fermat's Last Theorem; his 1837 theorem on primes in arithmetic progressions invented analytic number theory; and between those he quietly rebuilt the foundations under mathematical physics — the modern definition of a function is his, and so is the insistence that boundary-value problems be posed before being solved. The problem of finding a harmonic function from its boundary values acquired his name through his Berlin lectures, along with the variational "Dirichlet principle" (the harmonic solution minimizes $\int|\nabla V|^2$ ) that his student Riemann leaned on — and whose rescue, after Weierstrass punctured it, is a story for the next article. He married Rebecka Mendelssohn, sister of the composer; succeeded Gauss at Göttingen in 1855; and died four years later, having taught Riemann, Dedekind, and Kronecker what rigor sounded like. One datum per boundary point is a very Dirichlet kind of law: modest, exact, and non-negotiable.

History

1828

— Green's Essay shows boundary data plus interior equation determine the potential, founding the boundary-value viewpoint

1840

— Gauss recasts existence variationally in the Allgemeine Lehrsätze, ancestor of the Dirichlet principle

1850

— Dirichlet's Berlin lectures pose the boundary-value problem cleanly; his name attaches to the Dirichlet problem and principle

1877

— Carl Neumann's method of the arithmetic mean constructs solutions, giving the slope-data problem its name along the way

1902

— Hadamard defines well-posedness — existence, uniqueness, stability — and exhibits the Cauchy–Laplace catastrophe that makes the definition necessary

The physical picture

Think of the potential as a drumhead and the boundary as the rim where you are allowed to act. You can clamp the rim at chosen heights — that is Dirichlet, and the membrane interpolates. You can instead tilt the rim at chosen slopes and let its height find itself — that is Neumann, and the membrane's overall altitude is yours to choose afterwards, which is exactly the floating constant. What you cannot do is clamp and tilt the same piece of rim: a membrane whose edge height is fixed already has its edge slope determined by the physics inside; dictating both is giving the drum two contradictory orders.

Electrostatics dresses the drumhead in hardware. A conductor wired to a battery is a Dirichlet clamp: whatever induced charge it takes, its surface will sit at the set voltage, and the charge is harvested afterwards from the solved potential's slope. An insulated surface through which no field line may pass — or a symmetry plane of the problem, through which none does pass — is a free Neumann edge, $\partial V/\partial n=0$ . A sheet of known charge density prescribes the slope jump, hence Neumann data. And for a charged gadget alone in the universe, infinity is the remaining boundary, with $V\to0$ as its Dirichlet clause; forget it and your series solutions sprout growing modes with no condition left to kill them. One realistic actor refuses both costumes: the isolated, floating conductor, connected to nothing. Its surface is an equipotential — but at an unknown value; its total charge is known — but its pointwise $\sigma$ is not. That hybrid datum ("constant $V$ , value to be determined, subject to $\oint\sigma\,\mathrm{d}a = Q$ ") is legitimate and ubiquitous, and proving it still pins a unique solution is precisely the second uniqueness theorem, the centerpiece of the next article. Here we legislate; there they prosecute.

The taxonomy, and the two catches

The three legal data types on a boundary piece $S_i$ :

\text{Dirichlet:}\quad V\big|_{S_i} = g, \qquad \text{Neumann:}\quad \frac{\partial V}{\partial n}\Big|_{S_i} = h, \qquad \text{Robin:}\quad \Big(\alpha V + \beta\,\frac{\partial V}{\partial n}\Big)\Big|_{S_i} = k \qquad (1)

where: $g, h, k$ — prescribed functions on the boundary piece; $\partial/\partial n$ — outward normal derivative. Dirichlet is the electrode's contract; Neumann, via $\sigma=-\varepsilon_0\,\partial V/\partial n$ (the conductor version was derived in Surface Charge and Field at a Conductor; the general interface rules in Boundary Conditions on E and V), is the charge-sheet's; Robin blends them and appears at lossy or reactive surfaces — semiconductor contacts, electrochemical interfaces — more than in vacuum electrostatics. Mixing types across pieces is routine: worked example 1 solves a Dirichlet-left, Neumann-right sandwich. Catch one: pure Neumann data cannot be chosen freely. Integrate Poisson's equation over the region and apply the divergence theorem:

\oint_{\partial\Omega} \frac{\partial V}{\partial n}\,\mathrm{d}a = \int_\Omega \nabla^2V\,\mathrm{d}\tau = -\frac{Q_{\text{enc}}}{\varepsilon_0} \qquad (2)

where: $Q_{\text{enc}}$ — total charge inside $\Omega$ . This is Gauss's law wearing potential-theory clothes: the prescribed slopes fix the total flux through the boundary, and flux is not negotiable — it must match the enclosed charge. Data violating (2) describe a physically impossible situation and the mathematics responds with no solution at all (worked example 2 audits a case). Catch two: even compatible pure-Neumann data leave one degree of freedom loose,

V \;\text{and}\; V + c \;\text{solve the same Neumann problem for any constant } c \qquad (3)

where: $c$ — any constant; adding it changes no derivative, hence no datum. Nothing is broken — the field is unique — but the potential's zero is a convention the data never chose. Every Neumann-only simulation pins one node to $0\,\mathrm{V}$ by hand for exactly this reason; physically it is the freedom of where you clip the reference lead.

Figure

Hadamard's catastrophe — invisible boundary ripples, megavolt interior fiction

Derivation AWhy one datum per point — and why Cauchy data explode

Two claims to earn. First, the budget: a second-order elliptic problem accepts exactly one condition per boundary point, so Dirichlet-plus-Neumann on the same piece overdetermines. Second, Hadamard's sharper indictment: the overdetermined problem is not merely fussy but unstable — Cauchy data for Laplace's equation amplify boundary noise exponentially with depth, so even when a solution exists it is numerically worthless. The budget argument is honest counting in one dimension; the instability is a two-line explicit family in two.

Step 1. Counting, 1D model: on $0\le x\le d$ , $\mathrm{d}^2V/\mathrm{d}x^2=0$ gives $V=ax+b$ — two constants. Two boundary points, one datum each (a value or a slope), is two equations: generically a unique solution. Now prescribe Cauchy data at $x=0$ : $V(0)=b$ and $V'(0)=a$ — both constants are spent at one end, and whatever was prescribed at $x=d$ can no longer be honored except by accident. One point ate the whole budget. Why this generalizes: a second-order equation propagates one value-like and one slope-like freedom per direction; a closed elliptic boundary touches every interior point "from all sides," so distributing one datum per boundary point exactly exhausts the freedom. (The rigorous version is the uniqueness theorem next door.)

Step 2. Hadamard's family, 2D: consider, in the half-strip $x\ge0$ , the functions

V_n(x,y) = \frac{1}{n}\,\sin(ny)\,\sinh(nx).

Check harmonicity directly: $\partial_x^2V_n = +n\,\sin(ny)\sinh(nx)$ while $\partial_y^2V_n = -n\,\sin(ny)\sinh(nx)$ — the curvatures cancel exactly, so $\nabla^2V_n=0$ for every $n$ .

Step 3. Read off its Cauchy data on the wall $x=0$ :

V_n(0,y) = 0, \qquad \frac{\partial V_n}{\partial x}(0,y) = \sin(ny).

The data are bounded by $1$ for every $n$ — a fixed-size ripple, ever finer in wavelength.

Step 4. March inward to depth $x=L$ :

\max_y |V_n(L,y)| = \frac{\sinh(nL)}{n} \approx \frac{e^{nL}}{2n} \to \infty \quad (n\to\infty).

Put numbers on it: a ripple with $n = 100\,\mathrm{m^{-1}}$ (6 cm wavelength) and unit slope amplitude, marched a mere $L = 20\,\mathrm{cm}$ inward, has grown by $\sinh(20)/100 \approx 2.4\times10^{6}$ . Measurement noise at the boundary becomes megavolt fiction in the interior.

Step 5. Draw the moral precisely. Nothing is wrong with Laplace's equation; what's wrong is the question. The same equation with one datum per point smooths, averages, and stabilizes (the mean-value machinery of the previous article). Asked to run as an initial-value problem — data marching inward from one wall — it runs its smoothing in reverse and exponentiates. Well-posedness (existence, uniqueness, and continuous dependence on data — Hadamard's 1902 triple) is a property of the equation-plus-data pairing, and (1) lists the pairings that work. Verified: Hadamard's classical example (1923 lectures), reproduced in Jackson §1.9's discussion of appropriate boundary conditions; the sinh growth factor recomputed above; the 1D counting checks against the coax solve of Laplace's Equation (E2), which spent exactly two constants on two conditions.

Go deeper · Derivation B · 2 minWhat Neumann data really are — the charge budget and the floating zero

Derivation BWhat Neumann data really are — the charge budget and the floating zero

Neumann conditions look like the poor cousin — who prescribes derivatives? — until translated into physics: they prescribe charge, and their two peculiarities (the compatibility constraint and the undetermined constant) are both familiar electrostatics in disguise. We derive the constraint from the divergence theorem, exhibit the floating constant honestly, and translate both back into laboratory language.

Step 1. Let $V$ solve $\nabla^2V=-\rho/\varepsilon_0$ in $\Omega$ with prescribed $\partial V/\partial n = h$ on all of $\partial\Omega$ . Integrate the equation over $\Omega$ and use the divergence theorem on $\nabla^2V = \nabla\cdot(\nabla V)$ :

\oint_{\partial\Omega} h\,\mathrm{d}a = \oint_{\partial\Omega}\nabla V\cdot\hat{\mathbf{n}}\,\mathrm{d}a = \int_\Omega \nabla^2V\,\mathrm{d}\tau = -\frac{1}{\varepsilon_0}\int_\Omega\rho\,\mathrm{d}\tau.

Why there is no wiggle room: the left side is fixed by the data, the right by the sources; they are equated by the equation itself. If $\oint h\,\mathrm{d}a \ne -Q_{\text{enc}}/\varepsilon_0$ , the assumption "a solution $V$ exists" has produced a false statement — so no solution exists. Existence, unusually, has a checkable precondition.

Step 2. Translate: $-\varepsilon_0 h$ is the surface charge density the boundary would carry if it were a conductor, and $-\varepsilon_0\oint h\,\mathrm{d}a$ is the total charge the boundary must supply to terminate the interior's field lines. The constraint is flux conservation — Gauss's law — not a new axiom. This is also why an insulated closed cavity ( $h=0$ everywhere) can only house zero net charge.

Step 3. The floating constant, by direct check: if $V$ solves the Neumann problem, so does $V+c$ — the constant has zero gradient, so it alters neither the equation nor any prescribed slope. Both catches together say Neumann data determine the field exactly and the potential up to its reference; that the field is genuinely unique (no non-constant difference survives) is proved with the energy identity in Uniqueness Theorems for the Potential — we don't poach it here.

Step 4. Laboratory dictionary, assembled: battery-wired electrode → Dirichlet. Sheet of known $\sigma$ → Neumann with $h$ jumping by $-\sigma/\varepsilon_0$ (E1's crossing rules). Symmetry plane → $h=0$ , free of charge and of cost — the trick that lets solvers compute half a capacitor and mirror it. Isolated floating conductor → neither: constant-but-unknown $V$ plus known total $Q$ , the hybrid the second uniqueness theorem legitimizes. Infinity around a localized system → Dirichlet's $V\to0$ . Verified: constraint (2) is Jackson Eq. 1.61 territory (Neumann problems and their solvability); the insulated-cavity corollary cross-checks against the shielding article's Gaussian-surface bookkeeping; worked example 2 runs the budget audit numerically.

Limiting cases

Problem posed	Data	Outcome
Interior, Dirichlet everywhere	$V$ on all of $\partial\Omega$	unique solution — the chapter's default
Interior, Neumann everywhere	$\partial V/\partial n$ with $\oint h = -Q/\varepsilon_0$	unique field; $V$ floats by a constant
Neumann data violating the budget	$\oint h \ne -Q/\varepsilon_0$	no solution exists
Mixed pieces (electrode + insulation)	Dirichlet on some pieces, Neumann on rest	unique solution, no floating constant if any Dirichlet piece exists
Cauchy data on one piece	$V$ and $\partial V/\partial n$ together	overdetermined; unstable (Hadamard)
Localized system, open space	sources + $V\to0$ at infinity	infinity is the missing boundary piece

Worked examples

1 · A mixed sandwich — Dirichlet left, Neumann right (confidence builder)

A vacuum gap spans $0\le x\le d$ with $d=5.0\,\mathrm{cm}$ . The left wall is an electrode clamped at $V(0)=200\,\mathrm{V}$ (Dirichlet). The right wall is a conductor known to carry induced surface density $\sigma = -2.0\,\mu\mathrm{C/m^2}$ (Neumann — its potential is not prescribed). Find $V(x)$ and the right wall's potential.

Step 1. Classify and count: 1D Laplace ⇒ $V = ax+b$ : two constants, and we hold exactly two data — one per wall. Well-posed.

Step 2. Spend the Neumann datum — and anchor the normal before touching the algebra, because the formula $\sigma = -\varepsilon_0\,\partial V/\partial n$ was derived with $\hat{\mathbf{n}}$ pointing out of the conductor, which here is the $-\hat{x}$ direction (into the gap). So

\sigma = -\varepsilon_0\,\frac{\partial V}{\partial(-x)}\bigg|_{x=d} = +\varepsilon_0\,V'(d) = \varepsilon_0\,a \quad\Longrightarrow\quad a = \frac{\sigma}{\varepsilon_0} = \frac{-2.0\times10^{-6}}{8.854\times10^{-12}} = -2.26\times10^{5}\,\mathrm{V/m}.

Sanity-check the sign physically: $E_x = -a = +2.26\times10^{5}\,\mathrm{V/m}$ points in $+\hat{x}$ , i.e. field lines run toward the negative sheet — correct. Had we lazily used $\hat{n}=+\hat{x}$ , the slope would have flipped and the answer's sign with it; misanchored normals are the leading cause of death in Neumann problems.

Step 3. Spend the Dirichlet datum: $b = V(0) = 200\,\mathrm{V}$ . So

V(x) = 200 - 2.26\times10^{5}\,x \quad[\mathrm{V}], \qquad E_x = +2.26\times10^{5}\,\mathrm{V/m}.

Step 4. The output that was never an input: $V(d) = 200 - (2.26\times10^{5})(0.050) = 200 - 11{,}300 = -1.11\times10^{4}\,\mathrm{V}$ . The right plate floats at $-11.1\,\mathrm{kV}$ — determined by the solution, not prescribed by us. That is mixed data working as designed: each wall contributes one fact, the equation supplies the rest.

2 · Auditing a Neumann budget (exam level)

A simulation intern models a sealed, charge-free cubical cavity of side $L=10\,\mathrm{cm}$ , prescribing insulation ( $\partial V/\partial n = 0$ ) on five faces and a uniform inward field on the sixth: $\partial V/\partial n = -E_0$ with $E_0 = 1000\,\mathrm{V/m}$ (flux entering). The solver diverges. Diagnose, then state two legal repairs and what each leaves undetermined.

Step 1. Run the compatibility audit, equation (2): the data integrate to

\oint \frac{\partial V}{\partial n}\,\mathrm{d}a = 5\times0 + (-E_0)L^2 = -(1000)(0.10)^2 = -10\;\mathrm{V\!\cdot\! m},

which must equal $-Q_{\text{enc}}/\varepsilon_0 = 0$ for a charge-free cavity. It doesn't. No solution exists — the solver isn't broken, the physics is: field lines pour in through one face and are given nowhere to terminate.

Step 2. Quantify the contradiction. Taking the data at face value, the implied enclosed charge is

Q = -\varepsilon_0\oint\frac{\partial V}{\partial n}\,\mathrm{d}a = -(8.854\times10^{-12})(-10\;\mathrm{V\!\cdot\!m}) = +8.9\times10^{-11}\,\mathrm{C} \approx +89\,\mathrm{pC}.

But pause on the sign: inward field lines terminate on negative charge — positive 89 pC is backwards. The audit has caught a second bug: the intern wrote $h=-E_0$ as if $h$ were the normal field component $E_n$ , whereas $h \equiv \partial V/\partial n = -E_n$ . An inward field means $E_n=-E_0$ , hence $h=+E_0$ , $\oint h\,\mathrm{d}a = +10\,\mathrm{V\!\cdot\!m}$ , and the honestly implied charge is $Q = -89\,\mathrm{pC}$ — negative, drinking the inward flux, as intuition demands. Either way the cavity was declared charge-free, so with $\pm89\,\mathrm{pC}\neq0$ the data are incompatible and no solution exists.

Step 3. Legal repair one: balance the budget with geometry — prescribe matching outward flux on the opposite face ( $h = -E_0$ there, net $\oint h = 0$ ). The cavity then supports the uniform field $\mathbf{E} = E_0\hat{x}$ , $V = -E_0x + c$ — with $c$ genuinely undetermined: pure Neumann data never choose the zero. Repair two: convert one face to Dirichlet (say $V=0$ ) — any budget then self-consistently adjusts, and the constant is pinned.

Step 4. Moral, at solver scale: check $\oint h\,\mathrm{d}a$ against $-Q/\varepsilon_0$ before iterating, mind whether $h$ means $\partial V/\partial n$ or $E_n$ (a sign that swaps the required charge from $-89$ to $+89\,\mathrm{pC}$ ), and pin one reference node in any all-Neumann run. Three lines of audit that save an afternoon.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §3.1 · Jackson, Classical Electrodynamics, 3rd ed., §1.8–1.10 · Zangwill, Modern Electrodynamics, §7.3–7.4 · Hadamard, Lectures on Cauchy's Problem (1923), Ch. 1.

Provenance. Equations are standard electrostatics. A verifier should: (1) re-derive the uniqueness identity from Green's first identity (Jackson §1.9, Griffiths §3.1.5); (2) confirm the Neumann consistency constraint from Gauss's law; (3) check the dielectric matching conditions against Griffiths' boundary-condition equations and Zangwill §6; (4) verify the line-charge field $E_s=\lambda/(2\pi\varepsilon_0 s)$ and its potential (Griffiths §2.3). No nonstandard numbers are used; $\varepsilon_0$ is the SI vacuum permittivity. Additional checks for this treatment: Hadamard's family $\tfrac1n\sin(ny)\sinh(nx)$ (harmonicity and growth $\sinh(nL)/n\approx2.4\times10^6$ at $n=100\,\mathrm{m^{-1}}$ , $L=0.2\,\mathrm{m}$ ) and the worked numbers $2.26\times10^5\,\mathrm{V/m}$ , $-11.1\,\mathrm{kV}$ , $89\,\mathrm{pC}$ .

Builds on: Boundary Conditions on E and V · Surface Charge and Field at a Conductor · Laplace's Equation (E2) · Poisson's Equation (E2)

Leads to: Uniqueness Theorems for the Potential · Method of Images · Separation of Variables in Cartesian Coordinates

What is the division of labor between this article and the E1 Boundary Conditions on E and V?

E1's article derives the crossing rules: how $\mathbf{E}$ and $V$ behave across a charged interface — normal jump $\sigma/\varepsilon_0$ , tangential continuity, $V$ continuous, slope kinked. Those are facts about the field at internal surfaces, true in every problem. This article legislates the data: what you may prescribe on the boundary of the region you're solving in, so that the boundary-value problem is well-posed. You use E1's rules to translate physics (a charge sheet, a conductor skin) into data; you use this article's taxonomy to check the data make a solvable problem.

Why does a wired conductor give Dirichlet data when I don't know its surface charge?

Because the battery enforces the value, not the slope. A conductor held by a voltage source is an equipotential at a known potential — whatever induced charge that requires, the source supplies through the wire. So the honest input is $V=V_0$ (Dirichlet); the surface charge is an output, harvested after solving via $\sigma=-\varepsilon_0\,\partial V/\partial n$ . Trying to input the charge too would be Cauchy data — and the charge you'd guess would almost surely contradict the solution.

How do I hand "infinity" to a solver that only has a finite mesh?

Three standard moves. Truncate far away and clamp the outer box to $V=0$ — cheap, with error falling as the box grows (the potential of a localized neutral-ish system dies at least like $1/r$ ). Use "infinite elements" or a coordinate-inversion layer that maps $r\in(R,\infty)$ into a finite shell. Or match the numerical solution at the box to the analytic far-field multipole form and let the box carry Robin-type data. All three are implementations of the same principle: infinity is a boundary piece and owes the problem one datum.

Where do Robin conditions actually show up? Vacuum electrostatics never seems to need them.

Correct — in pure vacuum-and-conductor problems they're rare. They appear the moment the boundary has response: a semiconductor contact where surface current is proportional to local over-potential, an electrode with a thin resistive or dielectric coating (the coating's drop makes $V + \lambda\,\partial V/\partial n$ the natural fixed quantity), impedance boundary conditions in AC and wave problems, and convective surfaces in the heat-conduction twin of this theory. Robin is the "thin physical layer, modeled as algebra" condition.

Is the floating constant in Neumann problems ever physically meaningful?

No — and that's the insight, not a defect. Only potential differences and gradients are measurable; the constant is where you clip the reference lead. The Neumann problem is honest about this: given only charge/flux information, the theory refuses to invent a zero. Dirichlet problems seem to escape because the electrode data include a reference implicitly (the battery's other terminal, or grounded infinity). When an all-Neumann simulation demands you "ground one node," it is asking you to make the same conventional choice the lab makes.

Why exactly is the inward-continuation problem so unstable, physically?

Because charge-free electrostatics is a smoothing theory in the stable direction. The potential deep in a region is a heavily averaged, blurred functional of its boundary values — fine boundary detail dies off like $e^{-n(\text{depth})}$ (a fact the separation-of-variables articles make quantitative for slots). Continuation inward from data is the inverse of that blur: it must re-sharpen what averaging erased, multiplying each fine-scale component by $e^{+n(\text{depth})}$ . Any noise sitting in the fine scales gets the same amplification. Deconvolution of an exponential blur is the physical content of Hadamard's algebra.

Do these rules change inside dielectrics or with several materials?

The taxonomy survives; the bookkeeping grows. With materials, the natural interior equation is $\nabla\cdot(\varepsilon\nabla V)=-\rho_f$ and internal interfaces acquire matching conditions ( $V$ continuous, $\varepsilon\,\partial V/\partial n$ jumping by $-\sigma_f$ — E1's rules upgraded, treated properly in the E3 chapter). Those interface conditions are not boundary data — you don't choose them, the materials enforce them — but they consume the same accounting: each region needs its one-datum-per-point on the true outer boundary, and the interfaces stitch regions together. Solvers assemble exactly this ledger.