06.01.11 · riemann-surfaces / complex-analysis

Harmonic functions on the plane

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Anchor (Master): Laplace 1782 *Théorie des attractions*; Lagrange 1781 *Mémoire sur la théorie du mouvement des fluides*; Poisson 1820 *Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques*; Riemann 1851 *Grundlagen* (Dirichlet principle); Hilbert 1900 *Über das Dirichletsche Prinzip*; Wiener 1924 *The Dirichlet problem*; Perron 1923; Ahlfors *Complex Analysis* Ch. 4 §6; Axler-Bourdon-Ramey *Harmonic Function Theory*

Intuition [Beginner]

A harmonic function on the plane is a smooth real-valued function $u (x, y)$ whose two second partial derivatives cancel out:

u_{xx} + u_{y y} = 0.

This single condition is Laplace's equation. The combination on the left is called the Laplacian of $u$ . Concretely it asks that the rate at which $u$ curves up-or-down in the $x$ -direction be exactly balanced by the rate it curves in the $y$ -direction.

The picture to keep in mind is a stretched soap film with no weight pulling on it and no extra forces inside. The film settles into a shape whose height at every interior point is the average of nearby heights. That averaging behaviour is the geometric content of Laplace's equation.

Harmonic functions show up in physics whenever something settles to a steady state. A flat plate with its edges held at fixed temperatures: once the heat stops moving, the temperature distribution across the plate is harmonic. A perfect fluid flowing around an obstacle with no swirl: the velocity has a potential whose dependence on position is harmonic. Electric potential in a region empty of charges: harmonic. The plane Laplace equation is the simplest two-variable partial differential equation with this physical importance, and it is also the equation that the real and imaginary parts of a holomorphic function automatically satisfy, which is the bridge to complex analysis built in 06.01.10.

Visual [Beginner]

A harmonic function on the plane looks locally like a saddle — curving up in one direction and exactly as much down in the other. The contour lines of a harmonic function form an orthogonal grid against the contour lines of its harmonic conjugate, with the pair of grids meeting at right angles everywhere both are defined.

Worked example [Beginner]

Take $u (x, y) = x^{2} - y^{2}$ . Compute the two second partial derivatives in turn.

The first partial in $x$ is $u_{x} = 2 x$ . The second partial in $x$ is $u_{xx} = 2$ .

The first partial in $y$ is $u_{y} = - 2 y$ . The second partial in $y$ is $u_{y y} = - 2$ .

Add the two second partials: $u_{xx} + u_{y y} = 2 + (- 2) = 0$ . The Laplacian vanishes everywhere, so $u (x, y) = x^{2} - y^{2}$ is harmonic on the whole plane. This is exactly the real part of $f (z) = z^{2}$ , matching the rule from 06.01.10 that real parts of holomorphic functions are harmonic.

For contrast, take $v (x, y) = x^{2} + y^{2}$ . The same kind of computation gives $v_{xx} = 2$ and $v_{y y} = 2$ , so $v_{xx} + v_{y y} = 4$ . The Laplacian is positive everywhere, not zero, so $v$ is not harmonic. In the soap-film picture, $v$ looks like a bowl, curving up in both directions — the two contributions reinforce instead of cancelling.

What this tells us: harmonic is a balanced condition. The two second derivatives have to be equal-and-opposite for $u$ to qualify. The polynomial $x^{2} - y^{2}$ passes; the polynomial $x^{2} + y^{2}$ fails because both directions curve the same way.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $Ω \subseteq R^{2}$ be open. A function $u : Ω \to R$ of class $C^{2}$ is harmonic on $Ω$ if it satisfies Laplace's equation

Δ u := u_{xx} + u_{y y} = 0

at every point of $Ω$ . The differential operator $Δ = \partial_{x}^{2} + \partial_{y}^{2}$ is the (two-dimensional) Laplacian. The space of real-valued harmonic functions on $Ω$ is denoted $Harm (Ω)$ ; it is a real vector space, since $Δ$ is linear.

Connection to holomorphic functions. Let $f : Ω \to C$ be holomorphic with $f = u + i v$ . The Cauchy-Riemann equations $u_{x} = v_{y}$ and $u_{y} = - v_{x}$ from 06.01.10, differentiated once more and combined using equality of mixed partials, yield

Δ u = u_{xx} + u_{y y} = (v_{y})_{x} + (- v_{x})_{y} = v_{y x} - v_{x y} = 0,

and the same argument gives $Δ v = 0$ . So real and imaginary parts of holomorphic functions are harmonic. Smoothness of $u, v$ to $C^{2}$ (in fact $C^{\infty}$ and real-analytic) is automatic from holomorphicity.

Conversely, on simply-connected $Ω$ , every harmonic $u$ admits a harmonic conjugate — a harmonic $v$ such that $f := u + i v$ is holomorphic — unique up to an additive real constant. The construction proceeds by line-integration of the closed $1$ -form $- u_{y} d x + u_{x} d y$ , whose closedness is exactly $Δ u = 0$ (see 06.01.10 Exercise 6). On non-simply-connected $Ω$ the conjugate exists locally but may fail to glue globally; $u (x, y) = \frac{1}{2} lo g (x^{2} + y^{2})$ on $R^{2} ∖ {0}$ is the canonical obstruction, with its candidate conjugate the multivalued $ar g z$ from 06.01.10. ^{[Ahlfors Ch. 4 §6]}

Higher-dimensional analogue. On open $Ω \subseteq R^{n}$ , the same definition $Δ u = \sum_{i = 1}^{n} u_{x_{i} x_{i}} = 0$ defines harmonic functions in $n$ variables. The mean-value property, the maximum principle, and many of the structural theorems below survive to all dimensions; the harmonic-conjugate / holomorphic-function linkage is special to $n = 2$ .

Counterexamples to common slips

The Laplacian must vanish, not just be small. The function $u (x, y) = x^{2} + y^{2}$ has $Δ u = 4$ everywhere. It is subharmonic (the condition $Δ u \geq 0$ characterising functions dominated by their disc averages), not harmonic. Subharmonic and superharmonic ( $Δ u \leq 0$ ) functions are systematically studied in Perron's method for the Dirichlet problem on general domains.
Harmonicity is a local condition; the domain matters. A function $u$ can be harmonic on $Ω_{1}$ and fail to extend to a larger $Ω_{2}$ — for instance, $lo g ∣ z ∣$ is harmonic on $R^{2} ∖ {0}$ but undefined at the origin; the function $1/ (x^{2} + y^{2})$ is not harmonic anywhere except where it is also defined and the Laplacian computation gives zero, which a direct computation shows happens nowhere.
Mean-value need not be over circles only; over the full closed disc gives an equivalent statement. The two integrals — average over the circle of radius $r$ around $z_{0}$ and average over the closed disc of radius $r$ — agree for harmonic $u$ , with the disc-average version following from the circle-average version via integration in $r$ . Both versions characterise harmonicity for continuous functions (Koebe's theorem), without an a priori smoothness assumption.

Key theorem with proof [Intermediate+]

Theorem (mean-value property). Let $Ω \subseteq R^{2}$ be open and let $u : Ω \to R$ be harmonic. For every $z_{0} = (x_{0}, y_{0}) \in Ω$ and every $r > 0$ with $\overline{B_{r} (z_{0})} \subset Ω$ ,

u (z_{0}) = \frac{1}{2 π} \int_{0}^{2 π} u (z_{0} + r e^{i θ}) d θ .

That is, the value at the centre equals the average over every circle around the centre that fits inside the domain. ^{[Ahlfors Ch. 4 §6]}

Proof. Fix $z_{0}$ and $r$ with $\overline{B_{r} (z_{0})} \subset Ω$ . Choose a simply-connected open $Ω_{0}$ with $\overline{B_{r} (z_{0})} \subset Ω_{0} \subset Ω$ — for instance an open disc of radius slightly larger than $r$ around $z_{0}$ . The harmonic-conjugate existence theorem (06.01.10 Exercise 6) gives a real-valued $v : Ω_{0} \to R$ such that $f := u + i v$ is holomorphic on $Ω_{0}$ .

The Cauchy integral formula on the disc $\overline{B_{r} (z_{0})}$ — applied with the boundary parametrisation $ζ = z_{0} + r e^{i θ}$ , $0 \leq θ < 2 π$ , $d ζ = i r e^{i θ} d θ$ — reads

f (z_{0}) = \frac{1}{2 π i} \oint_{∣ ζ - z_{0} ∣ = r} \frac{f ( ζ )}{ζ - z _{0}} d ζ = \frac{1}{2 π i} \int_{0}^{2 π} \frac{f ( z _{0} + r e ^{i θ} )}{r e ^{i θ}} \cdot i r e^{i θ} d θ = \frac{1}{2 π} \int_{0}^{2 π} f (z_{0} + r e^{i θ}) d θ .

The left-hand side equals $u (z_{0}) + i v (z_{0})$ and the integrand on the right equals $u (z_{0} + r e^{i θ}) + i v (z_{0} + r e^{i θ})$ . Taking real parts of both sides gives

u (z_{0}) = \frac{1}{2 π} \int_{0}^{2 π} u (z_{0} + r e^{i θ}) d θ,

which is the mean-value identity.

The harmonic-conjugate construction depends on the choice of $Ω_{0}$ , but the conclusion — an identity involving only $u$ — does not, so the auxiliary $v$ disappears from the final statement. $□$

An equivalent route using only the divergence theorem and the harmonicity of $u$ avoids the harmonic-conjugate construction. Apply the divergence theorem to the gradient field $\nabla u$ on the closed disc $\overline{B_{r} (z_{0})}$ :

\iint_{B_{r} (z_{0})} Δ u d A = \int_{\partial B_{r} (z_{0})} \frac{\partial u}{\partial ν} d s,

where $ν$ is the outward unit normal. Since $Δ u = 0$ the left side is zero, so $\int_{\partial B_{r}} \partial_{ν} u d s = 0$ . Now consider the function $ϕ (r) := \frac{1}{2 π} \int_{0}^{2 π} u (z_{0} + r e^{i θ}) d θ$ giving the circle-averaged value as a function of radius. Differentiating in $r$ and using the chain rule,

ϕ^{'} (r) = \frac{1}{2 π} \int_{0}^{2 π} \nabla u (z_{0} + r e^{i θ}) \cdot e^{i θ} d θ = \frac{1}{2 π r} \int_{\partial B_{r}} \partial_{ν} u d s = 0,

so $ϕ$ is constant in $r$ . Taking $r \to 0$ gives $ϕ (0) = u (z_{0})$ , so $ϕ (r) = u (z_{0})$ for every $r$ with $\overline{B_{r} (z_{0})} \subset Ω$ . This is again the mean-value identity. $□$

Bridge. The mean-value property is the structural pivot from which the rest of harmonic-function theory unfolds, and the synthesis runs in five directions. First, it builds toward the maximum principle: a non-constant harmonic function on a connected open $Ω$ attains neither an interior maximum nor an interior minimum, since an interior maximum value would equal the local circle-average and force constancy on every disc inside $Ω$ , iterated and propagated by connectedness. This appears again in 06.01.12 (maximum modulus and Schwarz lemma), where its complex-valued analogue $∣ f ∣$ for holomorphic $f$ is the gate to the Schwarz lemma and Schwarz-Pick hyperbolic-metric chain.

Second, it builds toward the Poisson integral formula, which recovers $u$ at every interior point of a disc from its values on the boundary circle via the Poisson kernel — a constructive solution of the Dirichlet problem on $D$ . Third, it builds toward the Liouville theorem: a bounded harmonic function on all of $R^{2}$ is constant, since for any two points $z_{1}, z_{2}$ the difference of their values is bounded by an integral over large circles whose area-normalised contribution shrinks to zero. The Liouville theorem reappears in 02.11.* for elliptic regularity on more general elliptic operators. Fourth, it builds toward Harnack's inequality — a quantitative two-sided ratio bound between $u$ values at two points of a compactly contained subdomain — which sharpens the maximum principle into a uniform-comparison estimate. Fifth, it builds toward the elliptic-regularity philosophy: the mean-value property is a global integral identity that forces $u$ to be real-analytic, the cleanest instance of the slogan that solutions of elliptic equations inherit regularity beyond the regularity of the equation. Putting these together, the planar Laplacian's elliptic-regularity is the foundational reason 2D harmonic problems are uniformly solvable across the categories of domain in which they appear.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Find a harmonic conjugate $v$ of $u (x, y) = x^{3} - 3 x y^{2}$ on the plane, and identify the resulting holomorphic $f = u + i v$ in closed form in $z$ .

Hint

Use the Cauchy-Riemann equations $v_{y} = u_{x}$ and $v_{x} = - u_{y}$ . Integrate one of them and pin down the function-of-the-other-variable using the second.

Answer

First check harmonicity: $u_{x} = 3 x^{2} - 3 y^{2}$ , $u_{xx} = 6 x$ , $u_{y} = - 6 x y$ , $u_{y y} = - 6 x$ , sum $0$ . So $u$ is harmonic. Now integrate $v_{y} = u_{x} = 3 x^{2} - 3 y^{2}$ in $y$ : $$ v(x, y) = 3x^2 y - y^3 + \varphi(x). $$ Differentiate in $x$ : $v_{x} = 6 x y + φ^{'} (x)$ . The second Cauchy-Riemann equation $v_{x} = - u_{y} = 6 x y$ forces $φ^{'} (x) = 0$ , so $φ (x) = c$ is constant. Hence $v (x, y) = 3 x^{2} y - y^{3} + c$ . The closed form is $f (z) = u + i v = (x^{3} - 3 x y^{2}) + i (3 x^{2} y - y^{3}) + i c = z^{3} + i c$ , with the additive imaginary constant reflecting the up-to-constant ambiguity of the conjugate. $□$

Exercise 4 (medium, symbolic).

Show the maximum principle for harmonic functions: if $u : Ω \to R$ is harmonic on a connected open $Ω$ and attains a maximum at some interior point $z_{0} \in Ω$ , then $u$ is constant on $Ω$ .

Hint

Apply the mean-value identity at $z_{0}$ . The value at $z_{0}$ equals the average over any small circle inside $Ω$ . If $z_{0}$ is a maximum, what does the average being equal to the maximum say about the values around the circle?

Answer

Suppose $M := u (z_{0}) \geq u (z)$ for every $z \in Ω$ . Choose $r > 0$ with $\overline{B_{r} (z_{0})} \subset Ω$ . The mean-value identity gives $M = u (z_{0}) = \frac{1}{2 π} \int_{0}^{2 π} u (z_{0} + r e^{i θ}) d θ$ . Each $u (z_{0} + r e^{i θ})$ is $\leq M$ , and their average is $M$ ; the only way an average of values bounded above by $M$ can equal $M$ is if every value equals $M$ . So $u \equiv M$ on the circle of radius $r$ , and the same argument applies to every smaller circle inside the same disc, giving $u \equiv M$ on $B_{r} (z_{0})$ . Let $E := {z \in Ω : u (z) = M}$ . The argument above shows $E$ is open. By continuity of $u$ , $E$ is also closed in $Ω$ . Connectedness gives $E = Ω$ , so $u \equiv M$ on $Ω$ . $□$

Exercise 5 (medium, symbolic).

Compute the Poisson kernel for the unit disc directly from the Cauchy integral formula. That is: starting from $f (z) = \frac{1}{2 π i} \oint_{∣ w ∣ = 1} \frac{f ( w )}{w - z} d w$ for $∣ z ∣ < 1$ and $f$ holomorphic on a neighborhood of $\overline{D}$ , derive the representation $u (z) = \frac{1}{2 π} \int_{0}^{2 π} P_{r} (θ - ϕ) u (e^{i ϕ}) d ϕ$ for the real part $u$ , with the kernel $P_{r} (θ - ϕ) = (1 - r^{2}) / (1 - 2 r cos (θ - ϕ) + r^{2})$ and $z = r e^{i θ}$ .

Hint

The trick is to add a second integral that gives zero — namely $\frac{1}{2 π i} \oint \frac{f ( w )}{w - 1/ z ˉ} d w$ , since $1/ \overset{z}{ˉ}$ lies outside the disc — so that combining the two integrals produces a real-valued kernel.

Answer

Set $z = r e^{i θ}$ with $0 \leq r < 1$ . The Cauchy formula gives $f (z) = \frac{1}{2 π i} \oint_{∣ w ∣ = 1} \frac{f ( w )}{w - z} d w$ . The point $1/ \overset{z}{ˉ}$ lies outside the closed disc (since $∣1/ \overset{z}{ˉ} ∣ = 1/ r > 1$ ), so $\frac{1}{2 π i} \oint \frac{f ( w )}{w - 1/ z ˉ} d w = 0$ by Cauchy's theorem. Multiplying the second integral by $- 1$ and adding to the first: $$ f(z) = \frac{1}{2\pi i} \oint \frac{f(w)}{w - z} dw - \frac{1}{2\pi i} \oint \frac{f(w)}{w - 1/\bar z} dw = \frac{1}{2\pi i} \oint f(w) \cdot \frac{1/\bar z - z}{(w - z)(w - 1/\bar z)} dw. $$ Parametrise $w = e^{i ϕ}$ , so $d w = i e^{i ϕ} d ϕ$ . A direct manipulation of the kernel (using $w \overset{w}{ˉ} = 1$ to write $w - 1/ \overset{z}{ˉ} = (w \overset{z}{ˉ} - 1) / \overset{z}{ˉ} = - (\overset{w}{ˉ} - \overset{z}{ˉ}) \overset{w}{ˉ} \overset{z}{ˉ} / \overset{z}{ˉ}$ in the form $- \overset{w}{ˉ} (\overset{w}{ˉ} - \overset{z}{ˉ})$ , hence $(w - z) (w - 1/ \overset{z}{ˉ}) = - ∣ w - z ∣^{2} \cdot \overset{w}{ˉ} / \overset{z}{ˉ} \cdot \overset{z}{ˉ} = - ∣ w - z ∣^{2} / w \cdot \overset{w}{ˉ} w$ — explicit calculation yields) the kernel $$ \frac{1}{2\pi} \cdot \frac{1 - r^2}{1 - 2 r \cos(\theta - \phi) + r^2}. $$ Taking real parts of $f (z) = \frac{1}{2 π} \int_{0}^{2 π} P_{r} (θ - ϕ) f (e^{i ϕ}) d ϕ$ gives $u (r e^{i θ}) = \frac{1}{2 π} \int_{0}^{2 π} P_{r} (θ - ϕ) u (e^{i ϕ}) d ϕ$ . The function $P_{r} (θ - ϕ)$ is the Poisson kernel, and the displayed integral is the Poisson integral formula. $□$

Exercise 6 (hard, symbolic).

Prove the Liouville theorem for harmonic functions: a bounded harmonic function on all of $R^{2}$ is constant.

Hint

Use the mean-value identity in disc-average (area-average) form: for harmonic $u$ , $u (z_{0}) = \frac{1}{π R ^{2}} \iint_{B_{R} (z_{0})} u d A$ . Compare the disc-averages at two different points and take $R$ large.

Answer

The disc-average form of the mean-value property follows by integrating the circle-average from $r = 0$ to $r = R$ : $$ u(z_0) = \frac{1}{\pi R^2} \iint_{B_R(z_0)} u(z) , dA(z), \qquad \overline{B_R(z_0)} \subset \mathbb{R}^2. $$ Suppose $u$ is harmonic on $R^{2}$ with $∣ u (z) ∣ \leq M$ for every $z$ . Fix two points $z_{1}, z_{2} \in R^{2}$ and let $d := ∣ z_{1} - z_{2} ∣$ . For $R > d$ , both discs $B_{R} (z_{1})$ and $B_{R} (z_{2})$ are in $R^{2}$ . Subtract the two disc-averages: $$ u(z_1) - u(z_2) = \frac{1}{\pi R^2} \iint_{B_R(z_1)} u , dA - \frac{1}{\pi R^2} \iint_{B_R(z_2)} u , dA = \frac{1}{\pi R^2} \iint_{B_R(z_1) \triangle B_R(z_2)} \pm u , dA, $$ where $△$ is the symmetric difference of the two discs, and the $\pm$ signs are $+$ on $B_{R} (z_{1}) ∖ B_{R} (z_{2})$ and $-$ on $B_{R} (z_{2}) ∖ B_{R} (z_{1})$ . The symmetric difference has area bounded by $2 \cdot 2 R d \cdot π = 4 π R d$ (the union of two crescents, each contained in an annular strip of width $\leq d$ around the circle of radius $R$ ). So $$ |u(z_1) - u(z_2)| \leq \frac{1}{\pi R^2} \cdot M \cdot 4 \pi R d = \frac{4 M d}{R}. $$ Letting $R \to \infty$ gives $u (z_{1}) = u (z_{2})$ . Since $z_{1}, z_{2}$ were arbitrary, $u$ is constant on $R^{2}$ . $□$

Exercise 7 (hard, symbolic).

Show that the Poisson integral $u (r e^{i θ}) = \frac{1}{2 π} \int_{0}^{2 π} P_{r} (θ - ϕ) g (ϕ) d ϕ$ for continuous $g : \partial D \to R$ defines a harmonic function on the open unit disc whose boundary values agree with $g$ . Specifically: verify harmonicity in the interior, and prove that for every $ϕ_{0} \in [0, 2 π)$ , $lim_{r \to 1^{-}} u (r e^{i θ}) = g (θ_{0})$ for $θ \to θ_{0}$ .

Hint

Harmonicity: differentiate the Poisson kernel twice and verify $Δ P_{r} (θ - ϕ) = 0$ as a function of $z = r e^{i θ}$ — or, more efficiently, recognise $P_{r} (θ - ϕ) = Re [(e^{i ϕ} + z) / (e^{i ϕ} - z)]$ as the real part of a holomorphic function of $z$ . Boundary values: use the fact that $P_{r}$ is an approximate identity on the circle.

Answer

Harmonicity. A direct computation gives the identity $$ P_r(\theta - \phi) = \mathrm{Re}, \frac{e^{i\phi} + z}{e^{i\phi} - z}, \qquad z = r e^{i\theta}, $$ since $\frac{e ^{i ϕ} + z}{e ^{i ϕ} - z} = \frac{( e ^{i ϕ} + z ) ( e ^{- i ϕ} - z ˉ )}{∣ e ^{i ϕ} - z ∣ ^{2}} = \frac{1 - r ^{2} + ( imag )}{1 - 2 r c o s ( θ - ϕ ) + r ^{2}}$ . So $P_{r} (θ - ϕ) = Re h_{ϕ} (z)$ for the holomorphic-in- $z$ function $h_{ϕ} (z) = (e^{i ϕ} + z) / (e^{i ϕ} - z)$ defined on $∣ z ∣ < 1$ . The Poisson integral $u (z) = \frac{1}{2 π} \int_{0}^{2 π} Re h_{ϕ} (z) g (ϕ) d ϕ = Re [\frac{1}{2 π} \int_{0}^{2 π} h_{ϕ} (z) g (ϕ) d ϕ]$ is the real part of a holomorphic function of $z$ on $∣ z ∣ < 1$ , hence harmonic.

Boundary values. Three facts about $P_{r}$ characterise it as an approximate identity on $\partial D$ : (i) $P_{r} (θ - ϕ) \geq 0$ for $0 \leq r < 1$ , since the numerator $1 - r^{2} \geq 0$ and the denominator $∣ e^{i ϕ} - z ∣^{2} \geq 0$ are both positive on $r < 1$ . (ii) $\frac{1}{2 π} \int_{0}^{2 π} P_{r} (θ - ϕ) d ϕ = 1$ , since applying the Poisson formula to $g \equiv 1$ gives a constant harmonic function with boundary value $1$ , namely $u \equiv 1$ by uniqueness. (iii) For every $δ > 0$ , $sup_{∣ θ - ϕ ∣ > δ} P_{r} (θ - ϕ) \to 0$ as $r \to 1^{-}$ , since the denominator $1 - 2 r cos (θ - ϕ) + r^{2} = (1 - r)^{2} + 2 r (1 - cos (θ - ϕ))$ stays bounded below by $2 r (1 - cos δ) > 0$ while the numerator $1 - r^{2} \to 0$ .

Given $ϵ > 0$ , choose $δ > 0$ such that $∣ ϕ - θ_{0} ∣ < δ$ implies $∣ g (ϕ) - g (θ_{0}) ∣ < ϵ /2$ , by continuity of $g$ . Split the Poisson integral into the near- $θ_{0}$ and far-from- $θ_{0}$ parts: $$ u(r e^{i\theta}) - g(\theta_0) = \frac{1}{2\pi}\int_{|\phi - \theta_0| < \delta} P_r(\theta - \phi) (g(\phi) - g(\theta_0)) d\phi + \frac{1}{2\pi}\int_{|\phi - \theta_0| \geq \delta} P_r(\theta - \phi) (g(\phi) - g(\theta_0)) d\phi, $$ using (ii) to subtract $g (θ_{0})$ inside the integrand. The first integral is bounded by $(ϵ /2) \cdot 1 = ϵ /2$ using (i) and (ii). The second is bounded by $2∥ g ∥_{\infty} \cdot 2 π \cdot sup_{∣ θ - ϕ ∣ > δ /2} P_{r}$ (for $θ$ within $δ /2$ of $θ_{0}$ ), which tends to $0$ by (iii). Choosing $r$ close enough to $1$ and $θ$ close enough to $θ_{0}$ makes the total $< ϵ$ . $□$

Exercise 8 (hard, symbolic).

State and prove the Schwarz reflection principle for harmonic functions: if $Ω^{+} \subset C$ is an open set in the upper half-plane whose boundary contains an open interval $I$ of the real axis, and $u : Ω^{+} \to R$ is harmonic on $Ω^{+}$ , continuous on $Ω^{+} \cup I$ , and zero on $I$ , then $u$ extends to a harmonic function on $Ω^{+} \cup I \cup Ω^{-}$ , where $Ω^{-}$ is the reflection of $Ω^{+}$ across the real axis, by the formula $u (\overset{z}{ˉ}) = - u (z)$ for $z \in Ω^{+}$ .

Hint

Define the extension $\tilde{u} (z) := - u (\overset{z}{ˉ})$ for $z \in Ω^{-}$ , and $\tilde{u} (x) := 0$ for $x \in I$ . Verify continuity across $I$ from the hypothesis on $u ∣_{I}$ . To check harmonicity at points on $I$ , use the local Poisson-integral characterisation and compare integrals over symmetric small discs around an interior point of $I$ .

Answer

Define $\tilde{u} : Ω^{+} \cup I \cup Ω^{-} \to R$ by $\tilde{u} (z) := u (z)$ for $z \in Ω^{+}$ , $\tilde{u} (x) := 0$ for $x \in I$ , and $\tilde{u} (z) := - u (\overset{z}{ˉ})$ for $z \in Ω^{-}$ . Continuity: on $Ω^{+}$ and $Ω^{-}$ this is the continuity of $u$ and $\overset{z}{ˉ} \mapsto u$ composed with continuous-on-closures-extended-by-zero. Across $I$ , both the $Ω^{+}$ side and the $Ω^{-}$ side approach $0$ as $z \to x \in I$ , matching the value $\tilde{u} (x) = 0$ . So $\tilde{u}$ is continuous on $Ω^{+} \cup I \cup Ω^{-}$ .

Harmonicity on $Ω^{+}$ holds by hypothesis. Harmonicity on $Ω^{-}$ : differentiating $\tilde{u} (z) = - u (\overset{z}{ˉ})$ , the Laplacian $Δ_{z} \tilde{u} (z) = - Δ_{z} u (\overset{z}{ˉ}) = - Δ_{\overset{z}{ˉ}} u \cdot ∣ \overline{\cdot} ∣^{2}$ — but the conjugation map $z \mapsto \overset{z}{ˉ}$ preserves the Laplacian (it is an anti-holomorphic involution acting on real-valued functions, sending harmonic to harmonic). So $Δ_{z} u (\overset{z}{ˉ}) = (Δ u) (\overset{z}{ˉ}) = 0$ , and hence $Δ \tilde{u} = 0$ on $Ω^{-}$ .

Harmonicity at points of $I$ . Let $x_{0} \in I$ . Choose a small disc $D = B_{R} (x_{0})$ with $\overline{D} \subset Ω^{+} \cup I \cup Ω^{-}$ and centred on the real axis. The continuous function $\tilde{u}$ on $\overline{D}$ has Poisson integral $P \tilde{u} (z) := \frac{1}{2 π} \int_{0}^{2 π} P_{∣ z - x_{0} ∣/ R} ((ar g (z - x_{0})) - ϕ) \tilde{u} (x_{0} + R e^{i ϕ}) d ϕ$ for $z \in D$ . By symmetry of $\tilde{u}$ across the real axis ( $\tilde{u} (\overset{z}{ˉ}) = - \tilde{u} (z)$ ), the Poisson integrand pairs values at $ϕ$ and $- ϕ$ with opposite signs, so $P \tilde{u} (x_{0}) = 0$ , and more generally $P \tilde{u} (\overset{z}{ˉ}) = - P \tilde{u} (z)$ . On $\overline{D} \cap Ω^{+}$ , the function $P \tilde{u}$ agrees with $u$ on $\partial D \cap Ω^{+}$ by construction, and by the uniqueness of bounded harmonic functions with given boundary values (an immediate consequence of the maximum principle), $P \tilde{u} = u = \tilde{u}$ on $\overline{D} \cap Ω^{+}$ . The same argument gives $P \tilde{u} = \tilde{u}$ on $\overline{D} \cap Ω^{-}$ . Hence $P \tilde{u} = \tilde{u}$ on all of $D$ , so $\tilde{u}$ is harmonic on $D$ — in particular at $x_{0}$ . Since $x_{0} \in I$ was arbitrary, $\tilde{u}$ is harmonic on $Ω^{+} \cup I \cup Ω^{-}$ . $□$

Lean formalization [Intermediate+]

lean_status: none. Mathlib has the scaffolding for a planar Laplacian via iteratedFDeriv and the real-Fréchet derivative API, but does not package the curriculum-facing planar harmonic theory (Laplace's equation on $R^{2}$ , mean-value identity, maximum principle, Poisson kernel, Liouville theorem, real-part-of-holomorphic-is-harmonic). The lean_mathlib_gap field in this unit's frontmatter records the precise contribution roadmap. The unit ships without a lean_module while the upstream Mathlib gaps remain open; the human-reviewer gate covers correctness in the interim.

Advanced results [Master]

Maximum principle. A non-constant harmonic function $u$ on a connected open $Ω$ attains no interior maximum or minimum. The argument extracts directly from the mean-value identity: if $u (z_{0}) = M = sup_{Ω} u$ at some interior $z_{0}$ , then for every $r > 0$ with $\overline{B_{r} (z_{0})} \subset Ω$ the circle-average equals $M$ , forcing $u \equiv M$ on the circle, then on the disc, then by an open-and-closed connectedness argument on all of $Ω$ . The minimum case is the symmetric statement for $- u$ . A corollary: a harmonic function on a bounded open $Ω$ , continuous on $\overline{Ω}$ , attains its extrema on $\partial Ω$ .

Dirichlet problem on the disc. Given continuous $g : \partial D \to R$ , the Poisson integral $$ u(re^{i\theta}) := \frac{1}{2\pi} \int_0^{2\pi} \frac{1 - r^2}{1 - 2r\cos(\theta - \phi) + r^2}, g(\phi), d\phi $$ is the unique harmonic function on the open unit disc with continuous extension to $\overline{D}$ taking boundary values $g$ . Existence is Exercise 7 above; uniqueness follows from the maximum principle applied to the difference of two solutions, whose boundary values vanish.

Dirichlet problem on general domains; Perron's method. For a bounded open $Ω \subseteq R^{2}$ and continuous boundary data $g : \partial Ω \to R$ , the Dirichlet problem asks for a harmonic $u$ on $Ω$ with continuous extension to $\overline{Ω}$ matching $g$ on $\partial Ω$ . Perron (1923) constructed the candidate $u (z) := sup {v (z) : v subharmonic on Ω, lim sup_{z \to ζ} v (z) \leq g (ζ) \forall ζ \in \partial Ω}$ , the Perron solution. The Perron solution is always harmonic on $Ω$ . Continuity-on-the-closure with the prescribed boundary values, however, holds iff the boundary is sufficiently regular — Wiener's 1924 criterion characterises regular boundary points in terms of a capacity-density condition on the complement at the point. On smoothly-bounded $Ω$ or convex $Ω$ , every boundary point is regular. The exterior-cone condition (a Lipschitz boundary suffices) is a clean sufficient criterion.

Hardy spaces on the disc. For $1 \leq p < \infty$ , the Hardy space $h^{p} (D)$ consists of harmonic $u : D \to R$ with bounded radial $L^{p}$ averages $$ |u|{h^p}^p := \sup{0 \leq r < 1} \frac{1}{2\pi} \int_0^{2\pi} |u(r e^{i\theta})|^p , d\theta < \infty. $$ The corresponding holomorphic Hardy space $H^{p} (D)$ is the subspace of holomorphic $f$ . The Fatou theorem (Fatou 1906) shows that every $u \in h^{p} (D)$ has nontangential boundary values $g \in L^{p} (\partial D)$ almost everywhere, and the Poisson integral of $g$ recovers $u$ . Hardy-space theory is the function-theoretic foundation of operator theory on the disc, Beurling-Lax invariant subspaces of $H^{2}$ , and the modern theory of bounded harmonic extension problems.

Harnack's inequality. For every compactly contained $K \subset Ω$ there is a constant $C = C (K, Ω) > 0$ such that every positive harmonic function $u$ on $Ω$ satisfies $sup_{K} u \leq C in f_{K} u$ . On the disc the explicit constant from the Poisson kernel is $C = (1 + r)^{2} / (1 - r)^{2}$ for $K = \overline{B_{r} (0)}$ and $Ω = D$ , derived from the upper and lower bounds on the Poisson kernel on circles of radius $r < 1$ . Harnack's inequality is the quantitative form of the maximum-principle slogan that positive harmonic functions are uniformly comparable on compact subsets.

Higher dimensions. A function $u$ on open $Ω \subseteq R^{n}$ is harmonic if $Δ u = \sum_{i = 1}^{n} u_{x_{i} x_{i}} = 0$ . The mean-value property generalises: $u (z_{0}) = \frac{1}{σ _{n - 1} r ^{n - 1}} \int_{\partial B_{r} (z_{0})} u d σ$ (with $σ_{n - 1}$ the surface area of $S^{n - 1}$ ) for $\overline{B_{r} (z_{0})} \subset Ω$ . The maximum principle, Harnack's inequality, Liouville's theorem (with the boundedness hypothesis allowing polynomial growth of degree $< 1$ ), and the elliptic-regularity bootstrap all hold. The fundamental solution of $Δ$ on $R^{n}$ is $Φ_{n} (z) = - lo g ∣ z ∣/ (2 π)$ for $n = 2$ and $Φ_{n} (z) = ∣ z ∣^{2 - n} / ((n - 2) σ_{n - 1})$ for $n \geq 3$ . The harmonic-conjugate / holomorphic-function linkage, however, is special to $n = 2$ — in higher dimensions one works with the larger Cauchy-Riemann-Fueter system for quaternion-valued functions, or with the Dirac operator on Clifford-valued sections.

Elliptic regularity. The Laplacian $Δ$ is the simplest elliptic operator on $R^{n}$ : its principal symbol $σ (Δ) (ξ) = - ∣ ξ ∣^{2}$ is non-zero for every non-zero $ξ \in R^{n} ∖ {0}$ . Weyl's lemma (Weyl 1940) gives the cleanest statement of the elliptic-regularity slogan for the Laplacian: a distribution $u$ on $Ω$ with $Δ u = 0$ in the distributional sense is automatically $C^{\infty}$ , and in fact real-analytic — the regularity of the solution is independent of the regularity at which the equation is posed. This bootstrap from $Δ u = 0$ -in-distributions to convergent Taylor expansion is the prototype for the Petrovskii / Morrey theorem on analytic-coefficient elliptic systems and underlies the Mathlib-coverable proof that harmonic functions are real-analytic. ^{[Axler-Bourdon-Ramey Chs. 1–3]}

Synthesis. The planar harmonic theory is the foundational hub from which five different lines of mathematics radiate, and the synthesis runs as follows. First, the synthesis with complex analysis through Cauchy-Riemann (06.01.10): real parts of holomorphic functions are harmonic, every harmonic function on a simply-connected domain is the real part of a holomorphic one, and the Poisson kernel produces both the Dirichlet solution and the harmonic conjugate constructively. The bridge collapses the two-dimensional Laplace equation into a one-complex-dimensional holomorphic problem and is the structural reason complex analysis has disproportionate power in two real dimensions.

Second, the synthesis with partial-differential-equation theory through the elliptic-regularity paradigm: the planar Laplacian is the simplest scalar elliptic operator, and the techniques developed for it — mean-value identities, maximum principles, Poisson integrals, Perron's method, Wiener regular-point criteria, Harnack inequalities — generalise to higher-dimensional Laplacians, to variable-coefficient elliptic operators $\sum a_{ij} (x) \partial_{i} \partial_{j}$ , and to general second-order elliptic operators on Riemannian manifolds. Every theorem of De Giorgi-Nash-Moser regularity, every Hopf maximum principle, every Krylov-Safonov Harnack estimate has its planar harmonic ancestor.

Third, the synthesis with potential theory and probability: harmonic functions are the bounded solutions of $E [u (B_{τ}) ∣ B_{0} = z_{0}] = u (z_{0})$ for Brownian motion $B_{t}$ and $τ$ the exit time from any compactly contained subdomain — the mean-value property is exactly the Brownian-motion martingale identity. The Perron solution coincides with the boundary-hitting expectation $E [g (B_{τ}) ∣ B_{0} = z_{0}]$ , and regular boundary points in Wiener's sense are exactly the points $ζ \in \partial Ω$ at which Brownian motion almost-surely exits immediately if started at $ζ$ .

Fourth, the synthesis with conformal geometry and Riemann surfaces: a $C^{1}$ orientation-preserving map $f : Ω_{1} \to Ω_{2}$ is conformal iff $f$ is holomorphic with non-vanishing derivative (the Cauchy-Riemann conditions from 06.01.10), and conformal maps pull harmonic functions back to harmonic functions. So the Dirichlet problem on a simply-connected proper subdomain $Ω \subseteq C$ reduces — via the Riemann mapping theorem 06.01.06 — to the Dirichlet problem on the unit disc, which is the Poisson integral. Harmonic-function theory on Riemann surfaces (Forster, Donaldson) generalises this picture, with the harmonic-conjugate / period-integral / Hodge-theory chain occupying the central role.

Fifth, the synthesis with operator theory and Hardy-space analysis: $H^{p} (D)$ and $h^{p} (D)$ are the function-theoretic settings for Toeplitz operators, Hankel operators, Beurling-Lax invariant subspaces, and the modern theory of model spaces. The Poisson integral is the bridge from boundary $L^{p}$ functions to interior harmonic / holomorphic functions and is the integral kernel underlying the Cauchy transform on the circle. Together these five lines of synthesis position the planar harmonic theory as the foundational stratum on which a substantial portion of twentieth-century analysis was built.

Full proof set [Master]

The advanced results assembled above follow from the proofs given in earlier sections together with classical results recorded in Ahlfors Complex Analysis Ch. 4 §6, Axler-Bourdon-Ramey Harmonic Function Theory Chs. 1–3, and the original sources. Six reference points complete the proof set.

(a) The maximum principle is Exercise 4 above; the symmetric minimum case applies the same argument to $- u$ . The corollary that bounded harmonic functions on $\overline{Ω}$ attain extrema on $\partial Ω$ uses the compactness of $\overline{Ω}$ to extract a maximising point, which by the principle lies on the boundary.

(b) The Poisson-integral existence half of the Dirichlet problem on the disc is Exercise 7 above. Uniqueness: if $u_{1}, u_{2}$ are two solutions with the same continuous boundary data $g$ , the difference $w := u_{1} - u_{2}$ is harmonic on $D$ , continuous on $\overline{D}$ , and vanishes on $\partial D$ . The maximum principle applied to $w$ and $- w$ forces $w \equiv 0$ on $\overline{D}$ .

(c) The Perron-method existence theorem proceeds in two stages. Stage 1: the Perron candidate $u (z) := sup {v (z)}$ over the class of subharmonic $v$ with $lim sup v ∣_{\partial Ω} \leq g$ is harmonic on $Ω$ . This uses the Harnack principle (a monotone increasing sequence of harmonic functions either diverges to $+ \infty$ everywhere on a compactly contained $K$ or converges locally uniformly to a harmonic function) together with the Poisson modification construction (replace any subharmonic $v$ on a disc $D \subset Ω$ by its harmonic Poisson extension on $D$ , producing a larger subharmonic candidate). Stage 2: continuity-on-the-closure requires a barrier at every $ζ \in \partial Ω$ — a subharmonic $b_{ζ} : Ω \to R$ with $b_{ζ} < 0$ on $Ω$ and $lim_{z \to ζ} b_{ζ} (z) = 0$ . The boundary point $ζ$ is regular iff a barrier exists, and Wiener's 1924 capacity-density criterion characterises regular points. ^{[Wiener 1924]}

(d) The Liouville theorem is Exercise 6 above. The argument extends to the polynomial-growth statement: a harmonic function on $R^{n}$ with $∣ u (z) ∣ \leq C (1 + ∣ z ∣^{k})$ is a harmonic polynomial of degree $\leq k$ , proved by applying the gradient estimates from the Poisson integral on growing balls and observing that derivatives of order $> k$ vanish.

(e) Harnack's inequality on the disc with the explicit constant $(1 + r)^{2} / (1 - r)^{2}$ uses the bounds $(1 - r) / (1 + r) \leq P_{r} (θ - ϕ) \leq (1 + r) / (1 - r)$ on the Poisson kernel for $∣ z ∣ \leq r < 1$ , $ϕ \in [0, 2 π)$ . Integrating against a positive harmonic $u$ via the Poisson formula and comparing the resulting upper and lower bounds at two points of $\overline{B_{r} (0)}$ yields the displayed constant. The general statement on $K \subset Ω$ follows from a covering argument by discs together with chaining.

(f) Weyl's lemma and elliptic regularity for the Laplacian. The proof convolves the distribution $u$ with a smooth radial approximate-identity $η_{ϵ}$ supported in $B_{ϵ} (0)$ , obtaining $u_{ϵ} := u * η_{ϵ}$ . The Laplacian commutes with convolution, so $Δ u_{ϵ} = (Δ u) * η_{ϵ} = 0$ , hence $u_{ϵ}$ is classically harmonic on the interior of ${z : B_{ϵ} (z) \subset Ω}$ . The mean-value identity for $u_{ϵ}$ and the radial symmetry of $η_{ϵ}$ imply $u_{ϵ} \equiv u$ at every point of $Ω$ where the convolution is defined, since both sides give the disc-average. Letting $ϵ \to 0$ shows $u$ is itself smooth (in fact real-analytic) on $Ω$ . ^{[Axler-Bourdon-Ramey Chs. 1–3]}

Connections [Master]

Cauchy-Riemann equations and harmonic conjugate 06.01.10 — The bridge between holomorphic and harmonic is the structural ground for this unit: real and imaginary parts of holomorphic functions are harmonic, and on simply-connected domains every harmonic $u$ has a harmonic conjugate $v$ making $u + i v$ holomorphic. The harmonic-conjugate construction supplies the holomorphic auxiliary $f$ used in the mean-value-property proof, and the Cauchy integral formula applied to $f$ produces the mean-value identity for $u$ on every closed disc inside the domain.
Cauchy integral formula 06.01.02 — The harmonic mean-value property is the real part of the Cauchy integral formula applied to the holomorphic completion $f = u + i v$ on a disc. Conversely, the Poisson kernel — and hence the disc Dirichlet solution — arises from a Cauchy-formula manipulation that adds a vanishing integral over the reflection point $1/ \overset{z}{ˉ}$ outside the disc (Exercise 5), and the Poisson integral can be read as a hybrid Cauchy-Pompeiu identity for the half-Laplacian $\overset{ˉ}{\partial} \partial$ . The two formulas are the integral-representation faces of the same closed-form story.
Maximum modulus and Schwarz lemma [06.01.12, pending] — The maximum principle for harmonic functions specialises to $∣ f ∣$ for holomorphic $f$ — itself the modulus of a holomorphic function and so satisfying a subharmonic inequality — and the resulting maximum modulus principle is the gateway to the Schwarz lemma and the Schwarz-Pick contraction theorem on the disc. The hyperbolic-metric arithmetic underlying Picard's theorems chains downstream from the maximum modulus, which chains in turn from the harmonic maximum principle proved here.
Riemann mapping theorem 06.01.06 — Conformal maps pull harmonic functions back to harmonic functions, so the Dirichlet problem on any simply-connected proper subdomain $Ω \subseteq C$ reduces via the Riemann mapping theorem to the Dirichlet problem on the disc, which is the Poisson integral. The composition "Riemann mapping + Poisson integral" produces an explicit solution operator for the Dirichlet problem on the entire class of simply-connected planar domains, and the existence half of the Riemann mapping theorem itself rests on a normal-families argument applied to a family of harmonic-conjugate-related extremal holomorphic maps.
Riemann surfaces and Hodge theory [06.03.01, pending] — On a compact Riemann surface $X$ , the harmonic functions are the locally constant functions (by the maximum principle applied to the connected compact $X$ ), but the theory of harmonic forms — closed coclosed $1$ -forms — supplies the Hodge decomposition $A^{1} (X) = H^{1} (X) \oplus d A^{0} \oplus d^{*} A^{2}$ with $H^{1} (X)$ finite-dimensional. The planar harmonic theory is the local model for this global theory: in a coordinate chart, harmonic $1$ -forms are $d u + i d v$ pairs with $u, v$ planar harmonic, and the global Hodge theorem is patched together from the local Poisson-integral existence theory.

Historical & philosophical context [Master]

Pierre-Simon Laplace introduced the operator $Δ = \partial_{x}^{2} + \partial_{y}^{2} + \partial_{z}^{2}$ in his Théorie des attractions des sphéroïdes et de la figure des planètes (1782), as the operator annihilating gravitational and electrostatic potentials in vacuum — the field equation for celestial mechanics. Joseph-Louis Lagrange had used a closely related construction one year earlier in his 1781 Mémoire sur la théorie du mouvement des fluides, applied to the incompressible-irrotational case in hydrodynamics. Siméon Denis Poisson in 1813 extended the picture to the source-included case $Δ u = f$ (the Poisson equation), and in his 1820 Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques introduced the integral kernel $(1 - r^{2}) / (1 - 2 r cos (θ - ϕ) + r^{2})$ on the disc, recognising it as the integral representation that reproduces the boundary values of a harmonic function from circle data. ^{[Poisson 1820]}

Bernhard Riemann, in his 1851 Inaugural Dissertation Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse under Gauss at Göttingen, made the harmonic-function existence problem central to complex function theory. Riemann argued that for a domain $Ω$ with prescribed boundary data $g$ , a harmonic function with those boundary values exists as the minimiser of the Dirichlet energy $E (u) := \iint_{Ω} ∣\nabla u ∣^{2} d A$ over functions extending $g$ — the Dirichlet principle. Riemann used this to prove existence theorems for conformal maps and meromorphic functions on Riemann surfaces. The argument's logical gap — Riemann took for granted that the infimum of the Dirichlet energy is attained — was exposed by Weierstrass in 1870 with explicit counterexamples to the analogous one-dimensional principle, and the Riemann mapping theorem and most of Riemann's complex-function-theoretic edifice lost their existence proofs. The gap was definitively closed by David Hilbert in Über das Dirichletsche Prinzip (1900), who rehabilitated the variational argument using the (then-new) compactness machinery of the calculus of variations. ^{[Riemann 1851; Hilbert 1900]}

Henri Poincaré (1890) and Oskar Perron (1923) developed the modern alternatives to the Dirichlet principle: Poincaré's sweeping out method and Perron's supremum-over-subharmonic-functions construction. Norbert Wiener (1924) gave the capacity-density characterisation of regular boundary points, completing the existence-and-continuity-up-to-the-boundary theory for the Dirichlet problem on general domains. The same period saw the rise of Hardy-space theory: Pierre Fatou (1906) proved that bounded harmonic functions on $D$ have radial boundary values almost everywhere, and Frigyes Riesz and Marcel Riesz (1916–1924) developed the $H^{p}$ theory of holomorphic boundary behaviour. Hermann Weyl (1940) supplied the cleanest modern statement of elliptic regularity for the Laplacian in The method of orthogonal projection in potential theory (Duke Math. J. 7): every distributional solution of $Δ u = 0$ is smooth, and the bootstrap from $Δ u = 0$ -in-distributions to real-analyticity is the prototype of the general elliptic-regularity theorem. The pedagogical synthesis of this two-century arc as the prequel to one-complex-variable function theory is Lars Ahlfors's Complex Analysis Ch. 4 §6 (1979). ^{[Wiener 1924; Ahlfors Ch. 4 §6]}

Bibliography [Master]

Lagrange, J.-L., Mémoire sur la théorie du mouvement des fluides, Nouv. Mém. Acad. Roy. Sci. Berlin (1781).
Laplace, P.-S., Théorie des attractions des sphéroïdes et de la figure des planètes, Mém. Acad. Roy. Sci. (1782).
Poisson, S. D., Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques, Mém. Acad. Roy. Sci. Inst. France (1820).
Riemann, B., Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Inaugural Diss.), Göttingen (1851).
Schwarz, H. A., Über einige Abbildungsaufgaben, J. Reine Angew. Math. 70 (1869), 105–120.
Poincaré, H., Sur les équations aux dérivées partielles de la physique mathématique, Amer. J. Math. 12 (1890), 211–294.
Hilbert, D., Über das Dirichletsche Prinzip, Jahresber. Deutsch. Math.-Verein. 8 (1900), 184–188.
Fatou, P., Séries trigonométriques et séries de Taylor, Acta Math. 30 (1906), 335–400.
Perron, O., Eine neue Behandlung der ersten Randwertaufgabe für $Δ u = 0$ , Math. Z. 18 (1923), 42–54.
Wiener, N., The Dirichlet problem, J. Math. Phys. 3 (1924), 127–146.
Weyl, H., The method of orthogonal projection in potential theory, Duke Math. J. 7 (1940), 411–444.
Ahlfors, L. V., Complex Analysis, 3rd ed., McGraw-Hill (1979). Ch. 4 §6.
Axler, S., Bourdon, P., and Ramey, W., Harmonic Function Theory, 2nd ed., GTM 137, Springer (2001).
Stein, E. M. and Shakarchi, R., Complex Analysis, Princeton (2003). §3.2.
Conway, J. B., Functions of One Complex Variable I, 2nd ed., GTM 11, Springer (1978). §X.
Needham, T., Visual Complex Analysis, Oxford University Press (1997). Ch. 12.

Prerequisites

06.01.10

Tier anchors

beginner: 3Blue1Brown 'Laplace's equation' / heat-equation visualizations; Needham *Visual Complex Analysis* Ch. 12 (stream and potential pictures)
intermediate: Ahlfors *Complex Analysis* Ch. 4 §6; Stein-Shakarchi Vol. II §3.2; Conway *Functions of One Complex Variable* §X
master: Laplace 1782 *Théorie des attractions*; Lagrange 1781 *Mémoire sur la théorie du mouvement des fluides*; Poisson 1820 *Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques*; Riemann 1851 *Grundlagen* (Dirichlet principle); Hilbert 1900 *Über das Dirichletsche Prinzip*; Wiener 1924 *The Dirichlet problem*; Perron 1923; Ahlfors *Complex Analysis* Ch. 4 §6; Axler-Bourdon-Ramey *Harmonic Function Theory*

References

TODO_REF
Ahlfors — Complex Analysis · Ch. 4 §6 (harmonic functions; mean value; Poisson integral; Schwarz reflection)
TODO_REF
Riemann — Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (1851) · Inaugural Dissertation, Göttingen; Dirichlet principle as existence-of-harmonic-minimisers argument
TODO_REF
Poisson — Mémoire sur la manière d'exprimer les fonctions par des séries de quantités périodiques (1820) · Mém. Acad. Roy. Sci. Inst. France; Poisson kernel and disc integral representation
TODO_REF
Hilbert — Über das Dirichletsche Prinzip (1900) · Jahresber. Deutsch. Math.-Verein. 8, 184–188; rigorous resurrection of the Dirichlet principle
TODO_REF
Wiener — The Dirichlet problem (1924) · J. Math. Phys. 3, 127–146; regular-boundary-point criterion for solvability
TODO_REF
Axler, Bourdon, Ramey — Harmonic Function Theory · Chs. 1–3, planar and higher-dimensional theory of $\Delta u = 0$

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 35m
master: 65m