Laplace Equations

Introduction

The heat equation and the wave equation were each introduced with a promise that the present page now redeems. The classical trichotomy of linear second-order PDEs (parabolic, hyperbolic, elliptic) has a third member, the Laplace equation, and we committed to developing it in a subsequent page.

To present that third member as another point on the same axis, however, would be misleading. Heat and wave are initial-value problems in time. At \(t = 0\) something is prescribed, and the equation propagates it forward, parabolically in the heat case and hyperbolically in the wave case. The Laplace equation has no time variable. The problem it solves is structurally different, and the time axis along which heat and wave were arrayed is replaced here by the spatial boundary. Given values of \(u\) on the boundary \(\partial\Omega\) of a spatial region, what is \(u\) inside?

A second characterization stands alongside the first. Setting the time derivative of the heat equation to zero, or all time derivatives of the wave equation to zero, in the versions of those equations posed in two spatial variables, reduces each equation to a single common form: \[ \Delta u = \partial_{xx} u + \partial_{yy} u = 0. \] The Laplace equation is the time-independent form of the other two classical PDEs. On a bounded region whose boundary values are held fixed in time, it is the steady state to which a heat distribution settles after long time, and the equation satisfied by a vibrating membrane at rest.

The same equation arises in physics whenever an equilibrium of a diffusive or elastic medium is sought: the electrostatic potential in a region free of charge, the small-deflection shape of a soap film stretched across a wire frame. The solutions of \(\Delta u = 0\), called harmonic functions, are the static, equilibrium analogues of the dynamical objects studied in the heat and wave pages.

A third characterization, the most analytic of the three, asks what controls the solution. For the heat equation on the real line with bounded initial data \(f\), the maximum principle gives, for the solution obtained by convolution with the heat kernel, \[ \inf f \leq u(x, t) \leq \sup f \] for every \(x\) and every \(t \gt 0\). The solution is bounded at every point and every time by the extremes of the initial data. For the wave equation, no such bound holds in general, since a nonzero initial velocity can carry the solution outside the range of the initial displacement. The conserved quantity is the global mechanical energy, an integral invariant rather than a sup/inf estimate. For the Laplace equation, as the present page will prove, a bound by the data extremes returns \[ \min_{\partial\Omega} u \leq u(x) \leq \max_{\partial\Omega} u \] but with the controlling data shifted from the time-zero slice to the spatial boundary.

The three classical PDEs answer two related questions. The first is which data determine the solution, and the answer is initial data for heat and wave, boundary data for Laplace. The second is what controlling quantity governs its size: pointwise sup/inf bound by the data extremes in the parabolic and elliptic cases, an integral energy invariant in the hyperbolic case.

Laplace Equation on the Disk

The first concrete setting in which we solve the Laplace equation is the unit disk \(D = \{(x, y) : x^2 + y^2 \lt 1\}\) with boundary \(\partial D\) the unit circle. The Dirichlet problem on \(D\) prescribes the values of \(u\) on \(\partial D\) and asks for the harmonic extension into the interior: \[ \begin{cases} \Delta u(x, y) = 0, & (x, y) \in D, \\\\ u(x, y) = f(x, y), & (x, y) \in \partial D, \end{cases} \] where \(f\) is a prescribed continuous function on the unit circle.

The structural parallel with the heat and wave equations on the bounded interval is direct. In each case a second-order linear PDE is supplemented with data on the boundary of the spatial domain. The difference is the absence of a time variable and the absence of initial data. What was a one-dimensional spatial domain \((0, L)\) with two endpoint conditions is now a two-dimensional spatial domain \(D\) with a one-dimensional boundary \(\partial D\) carrying a full continuum of conditions. The natural choice of coordinates is polar.

Polar Coordinates and Separation of Variables

Setting \(x = r\cos\theta\) and \(y = r\sin\theta\) with \(r \in [0, 1]\) and \(\theta \in [-\pi, \pi]\), a direct chain-rule computation rewrites the Laplacian in the form \[ \Delta u = \partial_{rr} u + \frac{1}{r}\partial_r u + \frac{1}{r^2}\partial_{\theta\theta} u. \] The Dirichlet problem becomes \[ \begin{cases} \partial_{rr} u + \dfrac{1}{r}\partial_r u + \dfrac{1}{r^2}\partial_{\theta\theta} u = 0, & 0 \lt r \lt 1,\ -\pi \lt \theta \leq \pi, \\\\ u(1, \theta) = f(\theta), & -\pi \lt \theta \leq \pi, \end{cases} \] where, with a slight abuse of notation, \(f(\theta)\) denotes the boundary datum as a function on the circle and is assumed continuous and \(2\pi\)-periodic in \(\theta\).

Following the technique used for the heat equation on the bounded interval, we seek separated solutions \[ u(r, \theta) = R(r)\, \Theta(\theta). \] Substituting into the polar Laplace equation and multiplying through by \(r^2 / \bigl(R(r)\,\Theta(\theta)\bigr)\), valid wherever neither factor vanishes, separates the variables: \[ \frac{r^2 R''(r) + r R'(r)}{R(r)} = -\frac{\Theta''(\theta)}{\Theta(\theta)}. \]

The left-hand side depends only on \(r\), the right only on \(\theta\). Both must equal a common constant, which we denote \(\lambda\). The Dirichlet problem on the disk thus splits into a pair of ordinary differential equations coupled through the spectral parameter \(\lambda\): \[ \begin{align*} \Theta''(\theta) + \lambda\, \Theta(\theta) &= 0, \\\\ r^2 R''(r) + r R'(r) - \lambda\, R(r) &= 0. \end{align*} \]

The Angular Eigenvalue Problem

The angular equation \(\Theta'' + \lambda \Theta = 0\) is the same eigenvalue ODE that arose in the heat equation on the bounded interval, but the boundary conditions are different. There \(\Theta\) was required to vanish at two endpoints of an interval. Here \(\theta\) parametrizes a closed curve, and \(\Theta\) must instead be \(2\pi\)-periodic: \[ \Theta(\theta + 2\pi) = \Theta(\theta) \quad \text{for all } \theta \in \mathbb{R}. \]

This periodicity replaces the case analysis on the sign of \(\lambda\) that the Dirichlet endpoint conditions required. For \(\lambda \lt 0\), the general solution \[ \Theta(\theta) = c_1 e^{\sqrt{-\lambda}\,\theta} + c_2 e^{-\sqrt{-\lambda}\,\theta} \] grows or decays exponentially and admits no nontrivial periodic representative. For \(\lambda = 0\), \[ \Theta(\theta) = c_1 + c_2\theta \] is periodic only when \(c_2 = 0\), giving the constant solution \(\Theta_0 = 1\). For \(\lambda \gt 0\), the general solution \[ \Theta(\theta) = c_1 \cos\sqrt{\lambda}\,\theta + c_2 \sin\sqrt{\lambda}\,\theta \] is \(2\pi\)-periodic if and only if \(\sqrt{\lambda}\) is a positive integer. The eigenvalues are therefore \[ \lambda_n = n^2, \quad n \in \{0, 1, 2, \ldots\}, \] with corresponding two-dimensional eigenspaces spanned by \(\{\cos n\theta, \sin n\theta\}\) for \(n \geq 1\) (or, in complex form, by \(e^{in\theta}\) and \(e^{-in\theta}\)) and one-dimensional for \(n = 0\) (spanned by the constant).

Combining the positive and negative integer indices into a single signed parameter, we record the angular eigenfunctions in complex form as \[ \Theta_n(\theta) = e^{-in\theta}, \quad n \in \mathbb{Z}. \]

The Radial Equation

With \(\lambda = n^2\) determined, the radial equation becomes \[ r^2 R''(r) + r R'(r) - n^2 R(r) = 0. \] This is the Cauchy-Euler equation, whose general solution is obtained by trying \(R(r) = r^\alpha\). Substitution yields \(\alpha(\alpha - 1) + \alpha - n^2 = \alpha^2 - n^2 = 0\), so \(\alpha = \pm n\) for \(n \geq 1\). The general solution is \[ R_n(r) = A_n\, r^{n} + B_n\, r^{-n} \quad (n \geq 1), \] with the degenerate case \(n = 0\) handled separately. The radial equation \(r^2 R'' + r R' = 0\) factors as \(r \cdot (r R')' = 0\), so on \(r \gt 0\) we have \((r R')' = 0\). This gives \(r R'(r) = B_0\) constant and hence \(R_0(r) = A_0 + B_0 \log r\).

In both cases, one of the two basis solutions is singular as \(r \to 0^+\). The factor \(r^{-n}\) blows up for \(n \geq 1\), and \(\log r\) diverges to \(-\infty\) for \(n = 0\). We seek bounded harmonic functions on the entire open disk, including the origin. This regularity requirement forces \(B_n = 0\) for every \(n \geq 0\), leaving \[ R_n(r) = A_n\, r^{|n|}, \quad n \in \mathbb{Z}, \] where the absolute value covers the negative-index case uniformly (the radial factor depends on \(|n|\) only, while the angular factor distinguishes \(n\) from \(-n\)).

The Series Solution

The product solutions \(R_n(r)\, \Theta_n(\theta) = r^{|n|} e^{-in\theta}\) are each harmonic on the disk. By superposition, any convergent series of the form \[ u(r, \theta) = \sum_{n \in \mathbb{Z}} c_n\, r^{|n|}\, e^{-in\theta} \] formally satisfies the Laplace equation. The remaining task is to choose the coefficients \(c_n\) so that the boundary condition \(u(1, \theta) = f(\theta)\) holds. Setting \(r = 1\) gives \[ f(\theta) = \sum_{n \in \mathbb{Z}} c_n\, e^{-in\theta}, \] which is precisely the complex Fourier series of the boundary datum \(f\). The coefficients are therefore the Fourier coefficients \[ c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(\phi)\, e^{+in\phi}\, d\phi. \]

The boundary datum is decomposed into angular modes \(e^{in\phi}\), and each mode is propagated into the interior by the radial damping factor \(r^{|n|}\), which for \(n \neq 0\) decreases as \(r\) moves inward from the boundary at \(r = 1\) toward the origin. This is the structural mechanism of the disk Dirichlet problem. High-frequency boundary oscillations are damped most strongly toward the interior, while low-frequency modes persist deeper into the disk. The factor \(r^{|n|}\) shrinks faster for larger \(|n|\).

The Poisson Integral Formula

Substituting the Fourier coefficients back into the series and interchanging the order of summation and integration, justified by the absolute convergence of \(\sum r^{|n|}\) for \(r \lt 1\), yields \[ \begin{align*} u(r, \theta) &= \sum_{n \in \mathbb{Z}} \left(\frac{1}{2\pi}\int_{-\pi}^{\pi} f(\phi)\, e^{+in\phi}\, d\phi\right) r^{|n|}\, e^{-in\theta} \\\\ &= \int_{-\pi}^{\pi} f(\phi) \left(\frac{1}{2\pi}\sum_{n \in \mathbb{Z}} r^{|n|}\, e^{-in(\theta - \phi)}\right) d\phi. \end{align*} \]

The inner sum is a function of \(r\) and \(\theta - \phi\) alone. Writing \(\psi = \theta - \phi\) and splitting at \(n = 0\), we obtain \[ \sum_{n \in \mathbb{Z}} r^{|n|}\, e^{-in\psi} = 1 + \sum_{n=1}^{\infty} r^n\, e^{-in\psi} + \sum_{n=1}^{\infty} r^n\, e^{in\psi}. \] Each tail is a geometric series in the ratio \(r e^{\mp i\psi}\), of modulus \(r \lt 1\). Summing, \[ \begin{align*} \sum_{n=1}^{\infty} \bigl(r e^{-i\psi}\bigr)^n &= \frac{r e^{-i\psi}}{1 - r e^{-i\psi}}, \\\\ \sum_{n=1}^{\infty} \bigl(r e^{i\psi}\bigr)^n &= \frac{r e^{i\psi}}{1 - r e^{i\psi}}. \end{align*} \]

Adding the two tails to \(1\), placing over a common denominator \((1 - r e^{-i\psi})(1 - r e^{i\psi}) = 1 - 2r\cos\psi + r^2\), and simplifying the numerator yields \[ \sum_{n \in \mathbb{Z}} r^{|n|}\, e^{-in\psi} = \frac{1 - r^2}{1 - 2r\cos\psi + r^2}. \] The right-hand side depends on \(\psi\) only through \(\cos\psi\), so the kernel is even in \(\psi\) and the sign in the exponent of the original sum did not affect the final closed form. Dividing by \(2\pi\), we obtain the kernel.

The Poisson kernel is positive on its domain (the denominator \(1 - 2r\cos\psi + r^2 = (1 - r)^2 + 2r(1 - \cos\psi) \geq (1 - r)^2 \gt 0\) is strictly positive, and the numerator \(1 - r^2 \gt 0\) for \(r \lt 1\)). A direct integration of its series representation against \(d\psi\) gives \(\int_{-\pi}^{\pi} P_r(\psi)\, d\psi = 1\) (only the \(n = 0\) term survives after integration). With the kernel identified, the series solution takes integral form.

The Poisson kernel \(P_r(\theta - \phi)\) belongs to a broader family of integral kernels that arise in the theory of linear elliptic boundary-value problems. The fundamental object of that theory is the Green's function \(G(x, y)\) for the Laplace operator on a domain \(\Omega\), defined as the solution to \(\Delta_x G(x, y) = \delta_y(x)\) with \(G(x, y) = 0\) for \(x \in \partial\Omega\). Here \(\delta_y\) denotes the Dirac delta concentrated at \(y\), so \(G(\cdot, y)\) is the response to a unit point source there. Integrating \(G(x, y)\) against a source term \(g(y)\) in \(y\) solves the inhomogeneous problem \(\Delta u = g\) with homogeneous boundary data.

The Poisson kernel is the companion of \(G\) for the complementary problem treated on this page, namely the homogeneous equation \(\Delta u = 0\) with inhomogeneous boundary data \(u|_{\partial\Omega} = f\). It is recovered from \(G\) by differentiating in \(y\) along the outward unit normal \(n_y\) at a boundary point \(y\), a step that uses the symmetry \(G(x, y) = G(y, x)\), which we state without proof: \[ P(x, y) = \frac{\partial G(x, y)}{\partial n_y} \text{ for } y \in \partial\Omega. \]

The two kernels are therefore distinct objects. The function \(G\) satisfies an inhomogeneous equation involving the Dirac delta and is not harmonic, while \(P\) is itself harmonic on the open domain. The Green's function for the disk and its derivation via the method of images, along with the full distributional theory underlying \(\Delta G = \delta_y\), belong to a more advanced treatment than the present one and are not developed here. We introduce the name now because the same structural pair will reappear in the next section in a different geometric setting. That pair is a boundary kernel on a domain, arising as the normal derivative of an underlying Green's function.

Laplace Equation on the Half-Plane

Replacing the unit disk by the upper half-plane \(\mathbb{H} = \{(x, y) : y \gt 0\}\) changes the analytic situation in the same structural way that passing from the bounded interval to the real line did for the heat equation. The boundary loses its compactness, the angular Fourier series of the disk becomes the continuous Fourier transform on \(\mathbb{R}\), and the discrete sum over modes is replaced by an integral over a continuous frequency variable. The Fourier transform developed earlier is the natural tool, and the parallel with the disk treatment will be exact. A Fourier-side computation that diagonalizes the spatial derivative will produce, after inversion, a Poisson integral representation with a kernel of different geometric form but the same structural role.

The Dirichlet problem on the upper half-plane is \[ \begin{cases} \Delta u(x, y) = 0, & (x, y) \in \mathbb{H}, \\\\ u(x, 0) = f(x), & x \in \mathbb{R}, \end{cases} \] with \(f\) a bounded continuous boundary datum on \(\mathbb{R}\) and \(u\) required to remain bounded on \(\mathbb{H}\). The Fourier derivation below additionally assumes decay and integrability in \(x\), \(L^1\) convergence to the boundary datum as \(y \to 0^+\), and the conditions as \(y \to \infty\) imposed in its theorem. Its final inversion step is formal, and the formula it produces is then verified directly for every bounded continuous \(f\).

The boundedness requirement picks out the natural function class for the problem, and some condition at infinity is indispensable. The function \(u(x, y) = y\) is harmonic and vanishes on the real line, so without such a condition, adding it to a solution produces a second solution with the same boundary datum, and uniqueness fails outright. Control at infinity plays here the role that regularity at the origin played on the disk.

Fourier Transform in the Horizontal Variable

Define the spatial Fourier transform of \(u(\cdot, y)\) for each fixed \(y \gt 0\): \[ \hat{u}(\xi, y) = \int_{-\infty}^{\infty} u(x, y)\, e^{ix\xi}\, dx. \] Under the transform, moving both \(x\)-derivatives onto the exponential, exactly as for the differentiation property, produces the factor \((-i\xi)^2 = -\xi^2\). Applying the transform to the Laplace equation therefore converts the partial differential equation in \((x, y)\) into an ordinary differential equation in \(y\) alone, with \(\xi\) entering as a parameter:

The structural parallel with the disk is exact. There the angular Fourier coefficients \(c_n\) of the boundary datum were propagated inward by the radial factor \(r^{|n|}\), which for \(n \neq 0\) decays as \(r\) moves from the boundary \(r = 1\) toward the interior origin \(r = 0\). Here the continuous Fourier transform \(\hat{f}(\xi)\) of the boundary datum is propagated upward by the factor \(e^{-|\xi| y}\), which for \(\xi \neq 0\) decays as \(y\) moves from the boundary \(y = 0\) into the interior \(y \gt 0\). In both cases, high-frequency boundary content is damped fastest, and only low-frequency content survives deep into the interior. The factor \(r^{|n|}\) shrinks rapidly for large \(|n|\), and \(e^{-|\xi| y}\) shrinks rapidly for large \(|\xi|\).

The Half-Plane Poisson Kernel

To recover \(u(x, y)\) we invert the Fourier transform of the right-hand side. Applying the Fourier inversion formula formally, since its proof covers Schwartz functions and \(u(\cdot, y)\) need not be one, we obtain \[ u(x, y) = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\xi)\, e^{-|\xi| y}\, e^{-ix\xi}\, d\xi. \]

The identification of this expression as a convolution of \(f\) with a kernel requires the inverse transform of \(e^{-|\xi| y}\) itself, which is computable in closed form by a direct split-integral computation. For fixed \(y \gt 0\), \[ \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-|\xi| y}\, e^{-ix\xi}\, d\xi = \frac{1}{2\pi}\left[\int_0^{\infty} e^{-\xi y - ix\xi}\, d\xi + \int_{-\infty}^{0} e^{\xi y - ix\xi}\, d\xi\right]. \] Each integral is elementary: the first equals \(\frac{1}{y + ix}\), the second \(\frac{1}{y - ix}\). Their sum is \[ \begin{align*} \frac{1}{y + ix} + \frac{1}{y - ix} &= \frac{(y - ix) + (y + ix)}{(y + ix)(y - ix)} \\\\ &= \frac{2y}{x^2 + y^2}. \end{align*} \] Dividing by \(2\pi\) gives the kernel.

The half-plane Poisson kernel is positive on its domain (\(y \gt 0\) makes both numerator and denominator positive), and a direct integration gives \[ \begin{align*} \int_{-\infty}^{\infty} P_y(x)\, dx &= \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{y}{x^2 + y^2}\, dx \\\\ &= \frac{1}{\pi}\left[\arctan\!\frac{x}{y}\right]_{-\infty}^{\infty} \\\\ &= 1. \end{align*} \] With the kernel identified, the inverse transform expresses the solution as a convolution. Since this route was formal, the theorem below verifies the resulting formula directly.

Parallel Structure: Disk and Half-Plane

The two Dirichlet problems just solved share more than terminology. In each case a Fourier-mode decomposition of the boundary datum (discrete on the circle, continuous on the line) is damped into the interior by a factor parametrized by the distance from the boundary, producing a positive harmonic kernel that integrates to one and concentrates onto a single boundary point. The disk kernel \[ P_r(\psi) = \tfrac{1}{2\pi}(1 - r^2)/(1 - 2r\cos\psi + r^2) \] and the half-plane kernel \[ P_y(x) = \tfrac{1}{\pi}\, y/(x^2 + y^2) \] are two realizations of this pattern, differing in geometric form but identical in role. Both arise as boundary normal derivatives of an underlying Green's function on their respective domains.

A further parallel reaches back to the heat and wave equations. The heat equation on the real line was solved by the heat kernel \[ K_t(x) = (4\pi k t)^{-1/2} e^{-x^2/(4kt)}, \] parametrized by the time variable \(t\), which concentrates onto the initial datum as \(t \to 0^+\) and provides the solution by convolution.

The wave equation also admits a convolution representation, but of a different kind. D'Alembert's formula convolves the initial displacement \(f\) with the pair of Dirac deltas \(\tfrac{1}{2}(\delta_{ct} + \delta_{-ct})\) and the initial velocity \(g\) with \(\tfrac{1}{2c}\mathbf{1}_{[-ct, ct]}\), where \(\mathbf{1}_{[-ct, ct]}\) is the indicator function of \([-ct, ct]\), equal to \(1\) on the interval and \(0\) off it. Neither kernel regularizes in the manner of the heat kernel. The pair of point masses transports \(f\) without gaining a derivative, and the box kernel gains exactly one derivative on \(g\), which only offsets the one-order gap between displacement and velocity data. Both kernels are supported in \([-ct, ct]\), which is finite propagation speed expressed at the level of kernels.

The Laplace equation on the disk and on the half-plane both admit Poisson-kernel representations parametrized by the boundary-distance coordinate, which plays in the elliptic setting the role that time played in the parabolic setting. Where heat has a one-parameter family of kernels concentrating onto the time-zero slice, Laplace has a one-parameter family of kernels concentrating onto the spatial boundary. Where heat propagates information forward in time by convolution with a Gaussian, Laplace propagates information inward from the boundary by convolution with a Poisson kernel.

Harmonic Function Properties

The Poisson integral formulas just constructed are tied to the disk and the half-plane, and to the coordinate systems adapted to those two domains. We now turn to two qualitative properties that do not depend on the shape of the domain. The mean value property holds for every harmonic function on any open subset of the plane, and the maximum principle holds on every bounded open set and, for bounded functions, on the half-plane as well. The disk Poisson integral already hints at both, since it writes \(u\) as a weighted average of boundary values, but neither proof depends on it.

The maximum principle in particular is the signature property of the elliptic case, the pointwise bound on the solution by its boundary data. It is the boundary-data counterpart of the initial-data bound proved in the heat-equation treatment, which mentions this elliptic form without proving it.

The Mean Value Property

The Poisson integral formula on the unit disk already suggests the statement of the mean value property. Evaluating the kernel at \(r = 0\) gives \(P_0(\psi) = \frac{1}{2\pi}\), independent of \(\psi\), so the Poisson representation of any harmonic function constructed via the Poisson integral collapses, at the center, to the plain average of its boundary values. This special-case identity furnishes the form of the mean value property. Asserting it for an arbitrary harmonic function on an arbitrary domain, rather than only for functions defined as Poisson integrals, requires a proof that does not presuppose the Poisson representation. The argument below uses the polar form of the Laplacian directly, avoiding any appeal to uniqueness of the Dirichlet problem (established below via the maximum principle).

The Maximum Principle

The mean value property has an immediate qualitative consequence. A harmonic function cannot attain a strict interior maximum on a connected domain. Any point at which the function takes its maximum value would be forced, by the average being equal to the maximum, to be a value that every nearby point on a surrounding circle also takes. Iterating this observation spreads the maximum value through the connected component. The precise statement is the elliptic counterpart of the heat-equation pointwise bound, with the initial-time data slice replaced by the spatial boundary.

The maximum principle yields uniqueness for the Dirichlet problem as an immediate corollary, paralleling the role energy conservation played for the wave equation. Suppose \(u_1\) and \(u_2\) are two solutions of the Dirichlet problem on a bounded \(\Omega\) with the same continuous boundary datum \(f\) on \(\partial\Omega\). Their difference \(w = u_1 - u_2\) is harmonic on \(\Omega\), continuous on \(\overline{\Omega}\), and satisfies \(w|_{\partial\Omega} = 0\). The maximum principle gives \(\max_{\overline{\Omega}} w = \max_{\partial\Omega} w = 0\) and \(\min_{\overline{\Omega}} w = \min_{\partial\Omega} w = 0\), so \(w \equiv 0\) on \(\overline{\Omega}\). Hence \(u_1 = u_2\).

The Poisson integral formula on the disk therefore gives the solution of the Dirichlet problem on the disk, not merely a solution. The bounded domain \(\Omega = D\) falls squarely under the corollary. The half-plane is unbounded, so the corollary does not apply to it directly, and without a condition at infinity uniqueness fails there, as the function \(u(x, y) = y\) already showed. For bounded functions, a comparison with a slowly growing harmonic function recovers the bound by boundary values.

Uniqueness on the half-plane now follows as on the disk. If \(u_1\) and \(u_2\) are bounded, continuous on \(\overline{\mathbb{H}}\), harmonic on \(\mathbb{H}\), and agree on \(\partial\mathbb{H}\), then \(w = u_1 - u_2\) satisfies the hypotheses of the theorem with \(w(t, 0) = 0\) for every \(t\), so \(w \equiv 0\). The half-plane Poisson integral is bounded, as shown in the proof of the Poisson integral formula on the half-plane, so no other bounded continuous extension solves the same Dirichlet problem, and the uniqueness claim of that formula is proved.

The maximum principle for bounded domains, together with its half-plane extension for bounded functions, discharges the marker placed in the introduction, where the boundary-data form was promised but not yet derived. The structural placement of the elliptic case alongside the parabolic pointwise bound and the hyperbolic energy invariant is codified in the section that follows.

The Algebraic Origin of the Trichotomy

The three classical PDEs developed on these pages share a common algebraic origin. A general second-order linear PDE in two variables has principal part \(A u_{xx} + B u_{xy} + C u_{yy}\), and its qualitative behavior at each point is governed by the discriminant \(B^2 - 4AC\). This is exactly the same expression that classifies plane conic sections \(A x^2 + B xy + C y^2 + \cdots = 0\). Negative discriminant gives ellipses and the elliptic Laplace equation \(\partial_{xx} u + \partial_{yy} u = 0\), whose discriminant equals \(-4\). Zero discriminant gives parabolas and the parabolic heat equation \(\partial_t u - k\, \partial_{xx} u = 0\), whose principal part \(-k\, \partial_{xx} u\) yields discriminant \(0\). Positive discriminant gives hyperbolas and the hyperbolic wave equation \(\partial_{tt} u - c^2\, \partial_{xx} u = 0\), whose discriminant equals \(4c^2\).

The discriminant is the surface manifestation of a deeper algebraic fact. The principal part \(A u_{xx} + B u_{xy} + C u_{yy}\) is the quadratic form associated with the symmetric coefficient matrix \[ M = \begin{pmatrix} A & B/2 \\\\ B/2 & C \end{pmatrix}, \] and \(B^2 - 4AC = -4 \det M\), so the discriminant records the sign of the determinant of \(M\), which in turn records the sign pattern of its eigenvalues.

Negative discriminant means \(\det M \gt 0\), so the two eigenvalues of \(M\) share the same sign. The matrix is then definite, and the PDE is elliptic. Positive discriminant means \(\det M \lt 0\), so the eigenvalues have opposite signs. The matrix is then indefinite, and the PDE is hyperbolic. Zero discriminant means \(\det M = 0\), so one eigenvalue vanishes. The matrix is then degenerate, and the PDE is parabolic.

The Laplace matrix \(\operatorname{diag}(1, 1)\) has eigenvalues \((+1, +1)\). The wave matrix \(\operatorname{diag}(-c^2, 1)\) (with the second variable identified with \(t\)) has eigenvalues \((-c^2, +1)\). The heat matrix \(\operatorname{diag}(-k, 0)\) has eigenvalues \((-k, 0)\). Naming the three cases elliptic, parabolic, and hyperbolic is not metaphor. The names record the eigenvalue sign pattern of a \(2 \times 2\) symmetric matrix, and the classification by eigenvalue signs extends to higher-dimensional PDEs, where the principal symbol is a larger symmetric matrix and the discriminant \(B^2 - 4AC\) gives way to its signature, which admits more sign patterns than these three.

The discriminant has a second meaning, geometric rather than algebraic. It governs the number of real solutions of the characteristic equation \[ A(dy/dx)^2 - B(dy/dx) + C = 0 \] for the level curves \[ \phi(x, y) = \text{const} \] along which information propagates. For the wave equation we identify the second variable \(y\) with time \(t\). The principal part \[ \partial_{tt} u - c^2 \partial_{xx} u \] gives \[ \begin{align*} A &= -c^2 \\\\ B &= 0 \\\\ C &= 1 \end{align*} \] and the characteristic equation \[ -c^2(dt/dx)^2 + 1 = 0 \] has the two real roots \[ \begin{align*} dt/dx &= \pm 1/c, \\\\ \text{or equivalently } dx/dt &= \pm c. \end{align*} \] These define two real characteristic families, the lines \[ x \pm ct = \text{const} \] along which d'Alembert's formula carried the traveling waves on the wave page.

For the heat equation the principal part is \(-\partial_{xx} u\), since the first-order term \(\partial_t u\) does not enter the classification. The quadratic in \(dt/dx\) is degenerate, and the characteristic condition reduces to \(\phi_x = 0\), giving a single family of characteristics \(t = \text{const}\). Information at any instant is coupled across the entire spatial axis at once. This is the infinite propagation speed of diffusion, recovered here as a geometric consequence of the parabolic signature.

For the Laplace equation, no real characteristics exist at all. The equation \(\phi_x^2 + \phi_y^2 = 0\) admits only the trivial solution over the reals. There is no curve along which information flows, no time direction, no notion of forward evolution.

The absence of real characteristics has a qualitative consequence. Singularities at the boundary have no channel along which to propagate into the interior. A jump in the initial displacement of the wave equation, inserted into d'Alembert's formula outside its classical hypothesis \(f \in C^2\), splits into two half-size jumps that travel along the two characteristic lines through the jump point. For the Laplace equation, harmonic functions are smooth, even real-analytic, at every interior point regardless of boundary regularity.

Results already proved on this page yield the smoothness part. The harmonicity proof for the Poisson integral shows that it is \(C^\infty\) inside \(D\). By the uniqueness corollary of the maximum principle, any function harmonic in \(D\) and continuous on \(\overline{D}\) equals the Poisson integral of its boundary values. Since translations and dilations preserve harmonicity, the same argument applies on every open disk whose closure lies in the domain. Every point of the domain lies in such a disk, so every harmonic function is \(C^\infty\) throughout its domain. Real-analyticity, the stronger part of the claim, belongs to elliptic regularity theory and is not developed here.

The three controlling quantities catalogued in the introduction are three faces of one algebraic fact, the eigenvalue sign pattern of the principal-part matrix. For the wave equation, two real characteristic families carry the data at finite speed, and the mechanical energy is conserved forward and backward in time. Degenerate-characteristic diffusion makes heat bounded pointwise by initial data through the maximum principle on the real line. No-characteristic ellipticity makes Laplace bounded pointwise by boundary data through the boundary maximum principle. The three classical PDEs realize three paradigms of information flow: harmonic interpolation, dissipation, and propagation. Each admits a discrete dual treated elsewhere on these pages. The graph Laplacian is the discrete Dirichlet energy operator, and the factorization \(\boldsymbol{L} = \boldsymbol{B}\boldsymbol{B}^\top\) in terms of the incidence matrix of a graph is the discrete counterpart of \(-\Delta\), the sign convention under which the continuous Laplacian becomes a non-negative operator.

The parallel running through the trichotomy is summarized below.

The sign pattern of two eigenvalues produces the analytic, geometric, and discrete structure of the entire trichotomy. Up to an overall sign and the order of the two eigenvalues, its three possible patterns for a nonzero \(M\) are \((+,+)\) (definite, elliptic), \((-,+)\) (indefinite, hyperbolic), and \((-,0)\) (degenerate, parabolic).