The maximum principle

Theorem

The maximum principle

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Statement

Let \( \Omega \subseteq \mathbb{R}^n \) be open, bounded and connected, and let \( u \in C^2(\Omega) \cap C(\overline{\Omega}) \) be harmonic in \( \Omega \), i.e. \( \Delta u = \sum_{i=1}^n \frac{\partial^2 u}{\partial x_i^2} = 0 \) at every point of \( \Omega \). Then (weak form) \( \displaystyle \max_{\overline{\Omega}} u = \max_{\partial \Omega} u \) and \( \displaystyle \min_{\overline{\Omega}} u = \min_{\partial \Omega} u \); and (strong form) if \( u \) attains its maximum or its minimum over \( \overline{\Omega} \) at some interior point \( x_0 \in \Omega \), then \( u \) is constant on \( \Omega \).

Why it matters

The maximum principle is the single structural fact that makes harmonic functions rigid and predictable: a solution of Laplace's equation cannot have a local bump or dip anywhere inside its domain, because a genuine interior extremum would force the function to be constant everywhere. Everything downstream — uniqueness for the Dirichlet problem, continuous dependence of solutions on boundary data, comparison arguments, Harnack's inequality, and the existence machinery of Perron's method — rests on this control.

Physically it encodes the diffusive character of the Laplacian: for a steady-state temperature distribution with no internal sources, heat cannot spontaneously accumulate to a peak in the interior; the hottest and coldest points must sit on the boundary where the temperature is imposed.

Hypotheses

\( \Omega \) is bounded. On the unbounded domain \( \Omega = \{ x_2 \gt 0 \} \subset \mathbb{R}^2 \), the function \( u(x_1,x_2) = x_2 \) is harmonic, vanishes on the boundary \( \{x_2=0\} \), yet is unbounded above in the interior — there is no boundary maximum to compare against because \( \overline{\Omega} \) is not compact.

\( u \) is continuous on \( \overline{\Omega} \) (so that \( \max_{\partial\Omega} u \) is meaningful and attained, and comparison with the boundary makes sense). Without boundary continuity one can redefine a harmonic \( u \) arbitrarily on \( \partial\Omega \) (e.g. set it to \( +100 \) at a single boundary point) without disturbing \( \Delta u = 0 \) inside; the "boundary maximum" then fails to control the interior at all.

\( u \) is (twice continuously differentiable and) harmonic throughout the open set \( \Omega \), not merely subharmonic or harmonic off a lower-dimensional exceptional set. The function \( u(x) = 1/|x| \) on the punctured ball \( \Omega = B_1(0)\setminus\{0\} \subset \mathbb{R}^3 \) is harmonic away from the origin but blows up as \( x \to 0 \); the single excluded interior point wrecks the conclusion because harmonicity fails to extend there.

\( \Omega \) is connected (the strong form only — the weak form needs no connectedness). On \( \Omega = B_1(1,0) \cup B_1(-1,0) \subset \mathbb{R}^2 \) (two disjoint disks), let \( u \equiv 0 \) on the left disk and \( u \equiv 5 \) on the right disk. Each piece is harmonic (constant), \( u \) attains its maximum \( 5 \) at every interior point of the right disk, yet \( u \) is not constant on all of \( \Omega \) — the strong principle's rigidity conclusion needs a single connected component to propagate through.

Proof

Mean value property: if \( u \) is harmonic in \( \Omega \) and \( \overline{B_r(x_0)} \subset \Omega \), then \[ u(x_0) = \frac{1}{|\partial B_r(x_0)|}\int_{\partial B_r(x_0)} u \, dS = \frac{1}{|B_r(x_0)|}\int_{B_r(x_0)} u \, dx. \]

Standard lemma (proved from Green's second identity applied to \( u \) and the fundamental solution, or equivalently by differentiating \( \phi(r) = \fint_{\partial B_r(x_0)} u\,dS \) and using \( \Delta u = 0 \) via the divergence theorem to show \( \phi'(r) \equiv 0 \), so \( \phi(r) = \phi(0^+) = u(x_0) \)); taken here as a cited prerequisite result, the surface and solid versions being equivalent by integrating in \( r \). B

Suppose \( u \) attains a maximum \( M = \max_{\overline{\Omega}} u \) at an interior point \( x_0 \in \Omega \). Fix any \( r \gt 0 \) with \( \overline{B_r(x_0)} \subset \Omega \) (possible since \( \Omega \) is open).

Definition of open set: every interior point has a ball neighbourhood contained in \( \Omega \). A

\[ 0 = M - u(x_0) = \fint_{B_r(x_0)} \big(M - u(x)\big)\, dx, \] using Step 1, and the integrand \( M - u \geq 0 \) everywhere on \( \overline{\Omega} \) since \( M \) is the global maximum.

Substitution of the mean value identity (Step 1) into \( M = u(x_0) \), then rearranged; non-negativity of the integrand is immediate from \( u(x) \leq M \) for all \( x \in \overline\Omega \supseteq B_r(x_0) \). C

A continuous, non-negative function whose integral over \( B_r(x_0) \) is zero must vanish identically on \( B_r(x_0) \); hence \( u \equiv M \) on \( B_r(x_0) \).

Standard measure-theoretic fact for continuous integrands: if \( g \geq 0 \) is continuous and \( g(y_1) \gt 0 \) at some point, continuity gives a neighbourhood where \( g \gt \tfrac12 g(y_1) \), forcing \( \int g \gt 0 \), a contradiction. Applied with \( g = M-u \), continuous since \( u \in C^2(\Omega) \). B

Let \( S = \{ x \in \Omega : u(x) = M \} \). Step 4 shows \( S \) is open (every point of \( S \) has a whole ball inside \( S \)); continuity of \( u \) shows \( S \) is relatively closed in \( \Omega \) (as the preimage of the closed set \( \{M\} \)).

Openness: repeat Steps 2–4 at any \( y \in S \) in place of \( x_0 \), since \( y \) is itself an interior maximiser (\( u(y)=M \) is the global max). Closedness: preimage of a closed set under a continuous map is closed in the domain — standard topology. B

Since \( \Omega \) is connected and \( S \subseteq \Omega \) is non-empty (\( x_0 \in S \)), open in \( \Omega \), and closed in \( \Omega \), it follows that \( S = \Omega \). Hence \( u \equiv M \) on \( \Omega \), proving the strong maximum principle.

Definition of connectedness: a connected space has no partition into two non-empty disjoint relatively open (equivalently, relatively clopen) subsets; since \( S \) and \( \Omega \setminus S \) would form such a partition unless \( \Omega\setminus S = \emptyset \). A

Weak form for the maximum: \( u \in C(\overline{\Omega}) \) and \( \overline{\Omega} \) is compact (closed and bounded, by hypothesis), so \( u \) attains its maximum \( M \) at some \( x^\ast \in \overline{\Omega} \). If \( x^\ast \in \Omega \), Step 6 gives \( u \equiv M \) on \( \Omega \), hence by continuity \( u \equiv M \) on \( \overline\Omega \), so in particular \( u = M \) somewhere on \( \partial \Omega \) too (as \( \partial\Omega \neq \emptyset \) for bounded \( \Omega \)). If \( x^\ast \in \partial\Omega \) already, there is nothing more to prove. Either way \( \max_{\overline\Omega} u = \max_{\partial\Omega} u \).

Extreme value theorem (continuous function on a compact set attains its extrema) plus the case split resolved by Step 6; boundary non-emptiness is a topological fact for bounded open sets in \( \mathbb{R}^n \). A

Weak and strong forms for the minimum follow by applying Steps 2–7 to \( -u \), which is harmonic since \( \Delta(-u) = -\Delta u = 0 \), and noting \( \min u = -\max(-u) \).

Linearity of \( \Delta \) and of "max/min under negation"; no new argument required. A

Result

u \text{ harmonic in bounded, connected, open } \Omega,\ u\in C^2(\Omega)\cap C(\overline\Omega) \implies \max_{\overline\Omega} u = \max_{\partial\Omega} u,\ \min_{\overline\Omega} u = \min_{\partial\Omega} u,

and any interior extremum forces \( u \) to be constant.

Reading. A harmonic function cannot manufacture a peak or a trough of its own inside its domain; whatever extreme values it reaches, it must reach them out on the boundary, and if it ever equals its extreme value even once in the interior, it is flat-out constant everywhere.

Scope. Holds for classical (\(C^2\)) harmonic functions on bounded, connected open subsets of \( \mathbb{R}^n \) for any \( n \geq 1 \), continuous up to the closure. It extends verbatim to subharmonic functions for the maximum half (\( \Delta u \geq 0 \Rightarrow \max_{\overline\Omega} u = \max_{\partial\Omega} u \)) and to superharmonic functions for the minimum half, and — via the same mean-value mechanism with parabolic balls — to caloric (heat-equation) functions with an appropriately restricted "parabolic boundary". It does not by itself extend to general second-order elliptic operators without extra sign/structure conditions (see Hopf's maximum principle for that generalisation), and it does not hold as stated for unbounded \( \Omega \) or for merely distributional solutions without continuity assumptions.

Corollaries & converses

Uniqueness for the Dirichlet problem: if \( u_1, u_2 \in C^2(\Omega)\cap C(\overline\Omega) \) are both harmonic with \( u_1 = u_2 \) on \( \partial\Omega \), then \( w = u_1-u_2 \) is harmonic with zero boundary data, so by the weak principle \( \max_{\overline\Omega} w = \min_{\overline\Omega} w = 0 \), giving \( u_1 \equiv u_2 \). Free consequence, no extra work.
Continuous dependence / stability: if \( |u_1-u_2| \leq \varepsilon \) on \( \partial\Omega \), the same argument applied to \( w=u_1-u_2 \) gives \( |u_1-u_2| \leq \varepsilon \) throughout \( \overline\Omega \), so the Dirichlet problem is well posed in the sup norm.
Comparison principle: if \( u,v \) are harmonic with \( u \leq v \) on \( \partial\Omega \), then \( u \leq v \) throughout \( \overline\Omega \) (apply the weak principle to \( u-v \)).
Converse fails: a function whose values are always sandwiched between its boundary max and min need not be harmonic — e.g. any constant-sign convex combination trick, or more simply any subharmonic non-harmonic function such as \( u(x)=|x|^2 \) on a ball in \( \mathbb{R}^n \) (harmonic only when \( n\Delta \)-trace vanishes, which \( |x|^2 \) does not satisfy since \( \Delta |x|^2 = 2n \neq 0\)) still obeys "max on boundary" for its maximum but violates it for the minimum, showing the one-sided bound alone does not characterise harmonicity.
The strong principle's rigidity ("attains interior extremum \(\Rightarrow\) constant") is genuinely an if-and-only-if in the trivial direction: constants trivially attain every value everywhere, but a non-constant harmonic function never attains an interior extremum — this is exactly the contrapositive proved above, not an independent fact.

Fails without

Drop boundedness of \( \Omega \): \( u(x,y)=x \) is harmonic on the whole plane, restricted to the half-plane \( \Omega=\{x\gt 0\} \) it vanishes nowhere on the boundary \( x=0 \) in a way that bounds it — indeed \( u \) is unbounded above on \( \Omega \) while \( u=0\) identically on \( \partial\Omega \); the equality \( \max_{\overline\Omega}u=\max_{\partial\Omega}u\) fails outright since the left side is \( +\infty \).
Drop harmonicity (allow \( \Delta u \gt 0\) somewhere, i.e. merely subharmonic without the full equality): on \( \Omega = B_1(0)\subset\mathbb{R}^2 \), \( u(x,y)=x^2+y^2 \) has \( \Delta u = 4 \gt 0 \) and indeed still satisfies the maximum half (\(\max\) on boundary), but its minimum \( 0 \) is attained at the interior point \( (0,0) \) without \( u \) being constant — the strong minimum principle fails for subharmonic (non-harmonic) functions, showing genuine harmonicity (not just an inequality) is needed for the full strong principle.
Drop connectedness of \( \Omega \): as in the Hypotheses section, a harmonic function taking different constant values on different connected components attains an interior extremum on one component without being globally constant.
Drop continuity up to \( \overline\Omega \): take \( \Omega = B_1(0)\subset\mathbb{R}^2 \) and \( u \) the harmonic function with boundary values equal to the Poisson kernel's approximate identity concentrating positive mass near a single boundary point but redefine \( u \) at that one boundary point to be \( -1000 \) (destroying continuity there); \( \max_{\partial\Omega}u\) as naively read off the (discontinuous) boundary data no longer controls \( \max_{\overline\Omega}u \) correctly.

Common errors

Concluding \( u \) is constant whenever it merely touches its boundary maximum value at some boundary point — the strong principle's rigidity requires an interior point; equality at the boundary is completely expected and gives no information.
Applying the principle to \( \Delta u = f \) with \( f \not\equiv 0 \) (Poisson's equation) as if it were still exactly harmonic — the maximum principle as stated needs \( \Delta u = 0 \) exactly; for \( \Delta u \geq 0 \) or \( \le 0\) only the one-sided (sub/superharmonic) version survives, with the direction of the inequality determining which extremum transfers to the boundary.
Forgetting the connectedness hypothesis and asserting global constancy from an interior extremum on a domain that is secretly disconnected (e.g. an annulus-like region that has been carelessly described as a single "domain" when it is really a union of components).
Trying to apply the mean value property (Step 1) on a ball that pokes outside \( \Omega \) — the identity requires \( \overline{B_r(x_0)}\subset\Omega \); using it on a ball that exits the domain silently imports boundary values incorrectly.
Treating "\(u\) achieves its max on the boundary" as meaning the max is achieved only on the boundary — for a non-constant harmonic function this is true, but students often phrase it as though interior values near the boundary can't equal the boundary max, which is false by continuity.

Discussion

The maximum principle is best understood as the elliptic shadow of a much larger family of comparison principles that pervade the theory of second-order PDE, from the heat equation's parabolic maximum principle to viscosity-solution theory for fully nonlinear elliptic and parabolic equations. Its proof mechanism — the mean value property forcing an interior extremum to "leak" into a whole neighbourhood via a zero-integral argument, then propagating that neighbourhood through connectedness — is a template reused throughout: wherever a mean-value-type identity or a comparison with a fundamental solution is available, some version of the maximum principle tends to follow.

Historically the principle for harmonic functions was already implicit in Gauss's work on potential theory in the 1830s via the mean value property, and it was placed on a fully rigorous footing alongside the systematic development of potential theory in the later nineteenth century. Eberhard Hopf's 1927 refinement — now called Hopf's lemma or the Hopf maximum principle — extended the strong principle to general uniformly elliptic operators \( Lu = a_{ij}\partial_{ij}u + b_i\partial_i u \) (with \( c \leq 0 \) for a zeroth-order term), using a barrier-function argument rather than the mean value property, since general elliptic operators have no exact mean value identity.

A useful companion fact, worth internalising alongside this theorem, is Harnack's inequality: for non-negative harmonic functions on a domain, the ratio \( \sup_K u / \inf_K u \) is bounded uniformly on compact \( K \Subset \Omega \) by a constant depending only on \( \Omega, K, n \). This is a quantitative strengthening in the same spirit — control of extremes by geometry — though it is a genuinely different (harder) statement than the maximum principle itself.

At a more structural level, the maximum principle is the reason the Dirichlet problem is amenable to Perron's method of subharmonic functions: one defines the solution as the supremum of all subharmonic functions lying below the boundary data, and the maximum principle (applied to sub/superharmonic comparison functions) is exactly what shows this supremum is itself harmonic and attains the correct boundary values at every regular boundary point. Common misconception: students often believe the maximum principle says a harmonic function's extrema are always attained on the boundary as opposed to possibly also in the interior — but the correct statement is an inequality of suprema (\( \max_{\overline\Omega}u = \max_{\partial\Omega}u \)), which is entirely compatible with a constant function attaining its "extremum" everywhere; the strong principle is precisely the clarification of when interior attainment can occur (only for constants).

Worked examples

Let \( \Omega = B_1(0) \subset \mathbb{R}^2 \) and let \( u \) be the harmonic function on \( \Omega \), continuous on \( \overline\Omega \), with boundary data \( u(\cos\theta,\sin\theta) = \sin(2\theta) \). Bound \( \|u\|_{L^\infty(\overline\Omega)} \) without solving for \( u \) explicitly.

Setup: existence of such \( u \) is granted (Poisson integral / Perron's method); we use only the maximum principle. A

On \( \partial\Omega \), \( |\sin(2\theta)| \leq 1 \) for all \( \theta \), with equality attained (e.g. at \( \theta = \pi/4 \)), so \( \max_{\partial\Omega} u = 1 \) and \( \min_{\partial\Omega} u = -1 \).

Elementary bound on \( \sin \) together with the extreme value theorem on the compact circle \( \partial\Omega \). A

By the weak maximum principle (Step 7 of the proof), \( \max_{\overline\Omega} u = \max_{\partial\Omega} u = 1 \) and \( \min_{\overline\Omega} u = \min_{\partial\Omega} u = -1 \), so \( -1 \leq u(x) \leq 1 \) for every \( x \in \overline\Omega \).

Direct application of the theorem's weak form. A

Since the boundary data \( \sin(2\theta) \) is non-constant, \( u \) is non-constant, so by the strong principle \( u \) cannot attain the value \( 1 \) or \( -1 \) at any interior point of \( B_1(0) \); the bound \( -1 \lt u(x) \lt 1 \) is in fact strict for \( x \in B_1(0) \).

Contrapositive of the strong maximum/minimum principle: interior attainment of an extremum forces constancy, and \( u \) is not constant. B

-1 \lt u(x) \lt 1 \ \ \forall x \in B_1(0), \qquad \|u\|_{L^\infty(\overline{B_1(0)})} = 1

The sup norm of the solution is pinned down exactly from the boundary data alone, with no need to compute the Poisson integral.

Prove uniqueness for the Dirichlet problem: if \( u \in C^2(\Omega)\cap C(\overline\Omega) \) solves \( \Delta u = 0 \) in \( \Omega \) (bounded, connected) with \( u = g \) on \( \partial\Omega \) for a given continuous \( g \), then \( u \) is the only such solution.

Setup for a comparison argument. A

Suppose \( u_1, u_2 \) both solve the problem. Set \( w = u_1 - u_2 \in C^2(\Omega)\cap C(\overline\Omega) \). Then \( \Delta w = \Delta u_1 - \Delta u_2 = 0 - 0 = 0 \), so \( w \) is harmonic, and \( w = g - g = 0 \) on \( \partial\Omega \).

Linearity of the Laplacian and of the boundary condition. A

By the weak maximum principle, \( \max_{\overline\Omega} w = \max_{\partial\Omega} w = 0 \); by the weak minimum principle, \( \min_{\overline\Omega} w = \min_{\partial\Omega} w = 0 \).

Direct application of the theorem (Step 7 and Step 8 of the proof) to \( w \). A

Hence \( 0 = \min_{\overline\Omega} w \leq w(x) \leq \max_{\overline\Omega} w = 0 \) for every \( x \in \overline\Omega \), so \( w \equiv 0 \), i.e. \( u_1 \equiv u_2 \) on \( \overline\Omega \).

Squeeze between equal bounds. A

u_1 \equiv u_2 \text{ on } \overline\Omega

The Dirichlet problem for Laplace's equation on a bounded connected domain has at most one classical solution, entirely as a consequence of the maximum principle applied to the difference of two candidate solutions.

Problems

Let \( \Omega = (0,1)\times(0,1) \subset \mathbb{R}^2 \) and let \( u \) be harmonic in \( \Omega \), continuous on \( \overline\Omega \), with \( u = 0 \) on the three sides \( x=0 \), \( x=1 \), \( y=0 \) and \( u = \sin(\pi x) \) on the side \( y=1 \). Show \( 0 \leq u \leq 1 \) throughout \( \overline\Omega \).
Solution
On \( \partial\Omega \), \( u \) takes the value \( 0 \) on three sides and \( \sin(\pi x) \) on the fourth, and \( 0 \leq \sin(\pi x) \leq 1 \) for \( x \in [0,1] \). Hence \( \max_{\partial\Omega} u = 1 \) (attained at \( x=1/2, y=1 \)) and \( \min_{\partial\Omega} u = 0 \). By the weak maximum principle, \( \max_{\overline\Omega} u = 1 \) and \( \min_{\overline\Omega} u = 0 \), so \( 0 \leq u(x,y) \leq 1 \) on all of \( \overline\Omega \).
Let \( u \) be harmonic on all of \( \mathbb{R}^2 \) and suppose \( u \) attains a global maximum at some point \( x_0 \in \mathbb{R}^2 \). Show \( u \) is constant. (Note \( \Omega=\mathbb{R}^2\) is unbounded, so the theorem as stated does not directly apply — justify carefully why the strong principle still works here.)
Solution
Apply the strong maximum principle on any bounded ball \( B_R(x_0) \) for arbitrary \( R \gt 0 \): \( u \) is harmonic on \( B_R(x_0) \) (connected, bounded, and \( u \in C^2 \cap C(\overline{B_R(x_0)}) \) since \( u \) is harmonic, hence smooth, on all of \( \mathbb{R}^2\)), and \( u(x_0) = \max_{\mathbb{R}^2} u \geq \max_{\overline{B_R(x_0)}} u\), so \( x_0\) is an interior maximiser of \( u \) on \( B_R(x_0)\). By the strong principle, \( u \) is constant on \( B_R(x_0) \). Since \( R \) was arbitrary, \( u \) is constant on all of \( \mathbb{R}^2\). (This shows the strong principle localises: it needs boundedness of the ball used in the argument, not of the ambient domain.)
Let \( u,v \) be harmonic on a bounded connected \( \Omega \), both in \( C^2(\Omega)\cap C(\overline\Omega) \), with \( u \leq v \) on \( \partial\Omega \) but \( u \not\equiv v \). Show \( u \lt v \) strictly at every interior point of \( \Omega \).
Solution
Let \( w = v - u \), harmonic (difference of harmonic functions), with \( w \geq 0 \) on \( \partial\Omega \). By the weak minimum principle, \( \min_{\overline\Omega} w = \min_{\partial\Omega} w \geq 0 \), so \( w \geq 0 \) throughout \( \overline\Omega \), giving \( u \leq v \) on \( \overline\Omega \). Suppose for contradiction \( w(x_1) = 0 \) for some interior \( x_1 \in \Omega \). Then \( x_1 \) is an interior point where \( w \) attains its minimum value \( 0 \) (since \( w \geq 0 \) everywhere and \( w(x_1)=0\)). By the strong minimum principle, \( w \) is constant, i.e. \( w \equiv 0 \), so \( u \equiv v \) — contradicting \( u \not\equiv v \). Hence \( w \gt 0 \), i.e. \( u \lt v \), at every interior point.
Let \( \Omega \subset \mathbb{R}^n \) be bounded, connected and open, and let \( u \) be harmonic in \( \Omega \), continuous on \( \overline\Omega \), with \( u \) non-constant. Show that \( u \) cannot attain a local (not just global) interior maximum, i.e. there is no interior point \( x_0 \) and radius \( \rho \gt 0 \) with \( \overline{B_\rho(x_0)} \subset \Omega \) such that \( u(x_0) \geq u(x) \) for all \( x \in B_\rho(x_0) \).
Solution
Suppose such \( x_0, \rho \) existed with \( u(x_0) \geq u(x) \) for all \( x \in B_\rho(x_0) \). Apply Steps 1–4 of the proof of the strong maximum principle verbatim, but restricted to the ball \( B_\rho(x_0)\): the mean value property (Step 1) still applies on any \( B_r(x_0) \) with \( r \leq \rho \) since \( \overline{B_r(x_0)}\subset \overline{B_\rho(x_0)}\subset\Omega\), and \( u(x_0)\) being the maximum of \( u \) over \( B_\rho(x_0) \) is exactly what Steps 2–4 need (they only ever use maximality over the ball in question, not global maximality over \( \overline\Omega \)). Hence \( u \equiv u(x_0) \) on \( B_\rho(x_0) \). Then the open/closed set argument of Steps 5–6 (with \( S = \{x\in\Omega : u(x) = u(x_0)\}\), now non-empty since it contains \( B_\rho(x_0)\)) shows \( S = \Omega \), so \( u \) is constant on \( \Omega \) — contradicting the hypothesis that \( u \) is non-constant. So no such local interior maximum can exist. (This shows the strong principle is really a statement about the impossibility of any interior local extremum, global or otherwise — the proof machinery never actually needed global maximality, only maximality over some ball.)
Construct an example showing the maximum principle's boundedness hypothesis cannot be weakened to "\( \Omega \) has finite Lebesgue measure" (as opposed to being bounded/compact-closure), by exhibiting an open connected set \( \Omega \subset \mathbb{R}^2 \) of finite measure and a non-constant harmonic \( u \in C^2(\Omega)\cap C(\overline\Omega) \) with \( u \) bounded on \( \partial\Omega \) but unbounded on \( \Omega \).
Solution
Take the cusp region \( \Omega = \{(x,y) : x \gt 1,\ 0 \lt y \lt 1/x^2\} \subset \mathbb{R}^2\). It is open, connected and unbounded (it extends to \( x \to \infty \)), yet has finite area, since \( |\Omega| = \int_1^\infty x^{-2}\,dx = 1 \lt \infty \). Let \( u(x,y) = x \); this is harmonic on all of \( \mathbb{R}^2 \) (\( \Delta u = 0 \) trivially), hence harmonic on \( \Omega \), and \( u \in C^2(\Omega)\cap C(\overline\Omega)\). But \( u \) is unbounded on \( \Omega \), since \( u(x,y) = x \to \infty \) as \( x \to \infty \) along, say, \( y = 1/(2x^2) \in \Omega \). This does not contradict the theorem: the theorem requires \( \Omega \) bounded (equivalently \( \overline\Omega \) compact), and finite Lebesgue measure does not imply boundedness — \( \Omega \) here is a genuine counterexample precisely because \( \overline\Omega \) fails to be compact, so \( \max_{\overline\Omega} u \) is not even attained and there is no boundary maximum for the interior values to be compared against. This shows boundedness (compactness of the closure), not merely finiteness of measure, is the hypothesis doing the work.