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 — given values of \(u\) on the boundary \(\partial\Omega\) of a spatial
region, what is \(u\) inside? — and the time axis along which heat and wave were
arrayed is replaced here by the spatial boundary.
A second characterization stands alongside the first. Setting the time derivative
of the heat equation to zero, or both time derivatives of the wave equation to
zero, reduces each equation to a single common form:
\[
\Delta u \;=\; \partial_{xx} u + \partial_{yy} u \;=\; 0.
\]
The Laplace equation is the time-evolution endpoint of the other two classical
PDEs — the steady state to which a heat distribution settles after long time,
and the equation satisfied by a vibrating string 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 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, the maximum principle gives
\[
\inf f \leq u(x, t) \leq \sup f
\]
for every \(x\) and every \(t > 0\):
the solution is bounded at every point and every time by the extremes of the
initial data. For the wave equation, no such \(L^\infty\) bound by the initial
data exists; 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: which data
determine the solution — initial data for heat and wave, boundary data
for Laplace — and 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 \(D = \{(x, y) : x^2 + y^2 < 1\}\) 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
\[
\begin{cases}
\partial_{rr} u + \dfrac{1}{r}\partial_r u + \dfrac{1}{r^2}\partial_{\theta\theta} u \;=\; 0,
& 0 < r < 1,\ -\pi < \theta \leq \pi, \\
u(1, \theta) \;=\; f(\theta), & -\pi < \theta \leq \pi,
\end{cases}
\]
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:
\[
\Theta(\theta + 2\pi) \;=\; \Theta(\theta)
\qquad \text{for all } \theta \in \mathbb{R}.
\]
This periodicity replaces the case analysis on the sign of \(\lambda\) that the
Dirichlet endpoint conditions required. For \(\lambda < 0\), 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 \(\lambda > 0\),
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
\[
\lambda_n \;=\; n^2, \qquad n \in \{0, 1, 2, \dots\},
\]
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
\[
\Theta_n(\theta) \;=\; e^{-in\theta}, \qquad n \in \mathbb{Z}.
\]
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
\[
R_n(r) \;=\; A_n\, r^{n} + B_n\, r^{-n}
\qquad (n \geq 1),
\]
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 \(r > 0\)
we have \((r R')' = 0\), giving \(r R'(r) = B_0\) constant and hence
\(R_0(r) = A_0 + B_0 \log r\). In both cases, one of the two basis solutions is
singular as \(r \to 0^+\) — \(r^{-n}\) 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
\[
R_n(r) \;=\; A_n\, r^{|n|}, \qquad n \in \mathbb{Z},
\]
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 decays geometrically 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 — the factor \(r^{|n|}\) shrinks faster for larger \(|n|\) — while
low-frequency modes persist deeper into the disk.
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 \(r < 1\) — 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 \(n = 0\),
\[
\sum_{n \in \mathbb{Z}} r^{|n|}\, e^{-in\psi}
\;=\; 1 + \sum_{n=1}^{\infty} r^n\, e^{-in\psi}
+ \sum_{n=1}^{\infty} r^n\, e^{in\psi},
\]
each tail is a geometric series in the ratio \(r e^{\mp i\psi}\), of modulus
\(r < 1\). Summing,
\[
\sum_{n=1}^{\infty} \bigl(r e^{-i\psi}\bigr)^n
\;=\; \frac{r e^{-i\psi}}{1 - r e^{-i\psi}},
\qquad
\sum_{n=1}^{\infty} \bigl(r e^{i\psi}\bigr)^n
\;=\; \frac{r e^{i\psi}}{1 - r e^{i\psi}}.
\]
Adding these 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.
Definition: Poisson Kernel on the Disk
For \(0 \leq r < 1\) and \(\psi \in \mathbb{R}\), the
Poisson kernel on the unit disk is
\[
P_r(\psi)
\;=\; \frac{1}{2\pi} \cdot \frac{1 - r^2}{1 - 2r\cos\psi + r^2}.
\]
Equivalently, \(P_r(\psi) = \frac{1}{2\pi}\sum_{n \in \mathbb{Z}} r^{|n|}\, e^{-in\psi}\)
as an absolutely convergent series.
The Poisson kernel is positive on its domain (the denominator
\(1 - 2r\cos\psi + r^2 = (1 - r)^2 + 2r(1 - \cos\psi) \geq (1 - r)^2 > 0\) is
strictly positive, and the numerator \(1 - r^2 > 0\) for \(r < 1\)), and 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.
Theorem: Poisson Integral Formula on the Disk
Let \(f\) be a continuous \(2\pi\)-periodic function on \(\mathbb{R}\)
(equivalently, a continuous function on \(\partial D\)). The function
\[
u(r, \theta)
\;=\; \int_{-\pi}^{\pi} P_r(\theta - \phi)\, f(\phi)\, d\phi
\qquad (0 \leq r < 1)
\]
is harmonic on \(D\), extends continuously to the closed disk
\(\overline{D}\), and satisfies \(u(1, \theta) = f(\theta)\) on
\(\partial D\). It is the unique such solution of the Dirichlet problem
on the disk; uniqueness is established later as a consequence of the
maximum principle.
Proof:
Harmonicity: we argue in two steps. First, for each fixed
\(\phi\), the kernel \(P_r(\theta - \phi)\) is itself harmonic as a function
of \((r, \theta)\) on the open disk. This follows from its series
representation
\[
P_r(\theta - \phi) = \frac{1}{2\pi}\sum_n r^{|n|} e^{-in(\theta - \phi)}
\]
: each term \(r^{|n|} e^{-in(\theta - \phi)}\) is one of the separated product
solutions constructed earlier and is therefore harmonic, and the series and
all its term-by-term partial derivatives in \((r, \theta)\) of any fixed
order converge absolutely and uniformly on each compact subset
\(K \subset D\) — a consequence of \(\sum |n|^k r^{|n|} < \infty\) for
\(r < 1\). Termwise application of \(\Delta\) therefore gives
\(\Delta P_r(\theta - \phi) = 0\) on \(D\).
Second, on each compact \(K \subset D\) the same termwise bounds make
\(P_r\) and its \((r,\theta)\)-second derivatives bounded uniformly in
\(\phi \in [-\pi, \pi]\). For continuous \(f\) on the compact interval
\([-\pi, \pi]\), differentiation under the integral sign is therefore valid
on \(K\), and
\[
\Delta u(r, \theta)
\;=\;
\int_{-\pi}^{\pi} \Delta_{(r,\theta)} P_r(\theta - \phi)\, f(\phi)\, d\phi
\;=\; 0
\]
on \(D\).
Boundary behaviour: as \(r \to 1^-\), the Poisson kernel
\(P_r(\psi)\) concentrates near \(\psi = 0\) — its peak value at \(\psi = 0\)
is \(\frac{1}{2\pi}\,\frac{1+r}{1-r}\), tending to \(+\infty\), while for each
fixed \(\delta > 0\) the denominator is bounded below by
\(1 - 2r\cos\delta + r^2 \to 2(1 - \cos\delta) > 0\) on
\(|\psi| \geq \delta\), so
\(\sup_{|\psi| \geq \delta} P_r(\psi) \to 0\) as \(r \to 1^-\). Combined
with positivity and the normalization \(\int_{-\pi}^{\pi} P_r(\psi)\, d\psi = 1\),
these three properties make \(\{P_r\}_{r < 1}\) an
approximate identity on the circle as \(r \to 1^-\).
For continuous \(f\), uniform continuity on the compact circle gives, for
each \(\varepsilon > 0\), a \(\delta > 0\) such that
\(|f(\phi) - f(\theta)| < \varepsilon\) whenever \(|\phi - \theta| < \delta\)
(taken mod \(2\pi\)). Using \(\int P_r = 1\) to write
\(u(r, \theta) - f(\theta) = \int_{-\pi}^{\pi} P_r(\theta - \phi)
\bigl[f(\phi) - f(\theta)\bigr]\, d\phi\), and splitting the integration
domain into \(|\theta - \phi| < \delta\) and \(|\theta - \phi| \geq \delta\)
(mod \(2\pi\)):
\[
\bigl|u(r, \theta) - f(\theta)\bigr|
\;\leq\;
\varepsilon \int_{|\theta - \phi| < \delta} P_r(\theta - \phi)\, d\phi
+ 2\|f\|_\infty \int_{|\theta - \phi| \geq \delta} P_r(\theta - \phi)\, d\phi.
\]
The first integral is bounded by \(\int P_r = 1\), giving an
\(\varepsilon\) contribution; the second is bounded by
\(2\pi \sup_{|\psi| \geq \delta} P_r(\psi)\), which can be made smaller than
\(\varepsilon\) by choosing \(r\) sufficiently close to \(1\). Both bounds
are uniform in \(\theta\), so
\(\|u(r, \cdot) - f\|_\infty \to 0\) as \(r \to 1^-\). This establishes
continuity of \(u\) on \(\overline{D}\) and the boundary value
\(u(1, \theta) = f(\theta)\).
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 \(\Delta_x G(x, y) = \delta_y(x)\) (the response to
a point source at \(y\)) with \(G(x, y) = 0\) for \(x \in \partial\Omega\). The
Green's function solves the inhomogeneous problem \(\Delta u = f\) with
homogeneous boundary data; the Poisson kernel is its companion for the
complementary problem treated on this page — the homogeneous equation
\(\Delta u = 0\) with inhomogeneous boundary data \(u|_{\partial\Omega} = f\) —
and is recovered from the Green's function by taking its normal derivative on
the boundary, \(P(x, y) = -\partial G(x, y)/\partial n_y\) for \(y \in
\partial\Omega\). The two kernels are therefore distinct objects: \(G\) satisfies
an inhomogeneous equation involving the Dirac delta and is not a harmonic
function, 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; the
name is introduced now because the same structural pair — a boundary kernel
on a domain, arising as the normal derivative of an underlying Green's
function — will reappear in the next section in a different geometric setting.
Laplace Equation on the Half-Plane
Replacing the unit disk by the upper half-plane \(\mathbb{H} = \{(x, y) : y > 0\}\)
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 continuous and suitably decaying boundary datum on \(\mathbb{R}\),
and \(u(x, y)\) required to decay as \(y \to \infty\) and to remain bounded as
\(|x| \to \infty\). These decay requirements pick out the natural function class
for the problem and rule out, for instance, the addition of a constant (which
trivially satisfies \(\Delta u = 0\) but fails the decay condition unless the
constant is zero) or polynomially growing harmonic backgrounds — most
pointedly, \(u(x, y) = y\) is itself a non-trivial harmonic function vanishing
on the real line, so without a growth restriction uniqueness fails outright;
the same role is
played here by decay-at-infinity that was played on the disk by regularity at
the origin.
Fourier Transform in the Horizontal Variable
Define the spatial Fourier transform of \(u(\cdot, y)\) for each fixed \(y > 0\):
\[
\hat{u}(\xi, y) \;=\; \int_{-\infty}^{\infty} u(x, y)\, e^{ix\xi}\, dx.
\]
Applying the transform to the Laplace equation and using the
differentiation property
twice in \(x\), which produces the factor \((-i\xi)^2 = -\xi^2\) converts the
partial differential equation in \((x, y)\) into an ordinary differential
equation in \(y\) alone, with \(\xi\) entering as a parameter:
Theorem: Fourier-Transformed Laplace Equation on the Half-Plane
Let \(u(x, y)\) be a \(C^2\) solution of the Laplace equation on
\(\mathbb{H}\), with \(u(\cdot, y)\) decaying at infinity in \(x\) for each
fixed \(y > 0\), and let \(\hat{u}(\xi, y)\) denote its spatial Fourier
transform. Then, for each \(\xi \in \mathbb{R}\), the function
\(y \mapsto \hat{u}(\xi, y)\) satisfies the second-order linear ODE
\[
\partial_{yy} \hat{u}(\xi, y) - \xi^2\, \hat{u}(\xi, y) \;=\; 0,
\qquad
\hat{u}(\xi, 0) \;=\; \hat{f}(\xi).
\]
Imposing additionally that \(\hat{u}(\xi, y) \to 0\) as \(y \to \infty\) for
each \(\xi \neq 0\), the unique solution is
\[
\hat{u}(\xi, y) \;=\; \hat{f}(\xi)\, e^{-|\xi| y}.
\]
Proof:
Differentiation under the integral sign in \(y\) is justified by the
regularity and decay hypotheses on \(u\). Then
\[
\begin{align*}
\partial_{yy} \hat{u}(\xi, y)
&= \int_{-\infty}^{\infty} \partial_{yy} u(x, y)\, e^{ix\xi}\, dx \\\\
&= -\int_{-\infty}^{\infty} \partial_{xx} u(x, y)\, e^{ix\xi}\, dx \\\\
&= -(-i\xi)^2\, \hat{u}(\xi, y) \\\\
&= \xi^2\, \hat{u}(\xi, y),
\end{align*}
\]
where the second equality uses the Laplace equation
\[
\partial_{yy} u = -\partial_{xx} u
\]
and the third applies the differentiation property of the Fourier transform twice in \(x\), with the
boundary terms from the two integrations by parts vanishing by the assumed
decay of \(u\) and \(\partial_x u\) in \(x\). The boundary condition
\(u(x, 0) = f(x)\) translates directly under the transform to
\(\hat{u}(\xi, 0) = \hat{f}(\xi)\).
For each fixed \(\xi\), the equation
\(\partial_{yy} \hat{u} = \xi^2\, \hat{u}\) is a scalar linear ODE in \(y\)
with characteristic equation \(\alpha^2 = \xi^2\), roots
\(\alpha = \pm |\xi|\). The general solution is
\[
\hat{u}(\xi, y) \;=\; A(\xi)\, e^{|\xi| y} + B(\xi)\, e^{-|\xi| y}.
\]
For \(\xi \neq 0\), the term \(e^{|\xi| y}\) grows without bound as
\(y \to \infty\); the decay requirement \(\hat{u}(\xi, y) \to 0\) forces
\(A(\xi) = 0\). The boundary condition then gives
\(B(\xi) = \hat{f}(\xi)\), yielding the stated solution. At \(\xi = 0\),
the ODE degenerates to \(\partial_{yy} \hat{u}(0, y) = 0\), whose bounded
solutions are constants; boundedness rules out the linear term, and the
boundary condition fixes the constant as \(\hat{f}(0)\), in exact agreement
with the limit \(\lim_{\xi \to 0} \hat{f}(\xi)\, e^{-|\xi| y} = \hat{f}(0)\).
The formula \(\hat{u}(\xi, y) = \hat{f}(\xi)\, e^{-|\xi| y}\) therefore
holds for all \(\xi \in \mathbb{R}\) by continuous extension.
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 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 decays as \(y\) moves from the boundary \(y = 0\) into
the interior \(y > 0\). In both cases, high-frequency boundary content is damped
fastest — \(r^{|n|}\) shrinks rapidly for large \(|n|\), and \(e^{-|\xi| y}\)
shrinks rapidly for large \(|\xi|\) — and only low-frequency content survives
deep into the interior. The damping factor in each case is parametrized by the
distance from the boundary, measured along the natural interior coordinate.
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:
\[
u(x, y)
\;=\; \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\xi)\, e^{-|\xi| y}\, e^{-ix\xi}\, d\xi.
\]
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 \(y > 0\),
\[
\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-|\xi| y}\, e^{-ix\xi}\, d\xi
\;=\; \frac{1}{2\pi}\left[\int_0^{\infty} e^{-\xi y - ix\xi}\, d\xi
+ \int_{-\infty}^{0} e^{\xi y - ix\xi}\, d\xi\right].
\]
Each integral is elementary: the first equals \(\frac{1}{y + ix}\), the second
\(\frac{1}{y - ix}\). Their sum is
\[
\frac{1}{y + ix} + \frac{1}{y - ix}
\;=\; \frac{(y - ix) + (y + ix)}{(y + ix)(y - ix)}
\;=\; \frac{2y}{x^2 + y^2}.
\]
Dividing by \(2\pi\) gives the kernel.
Definition: Poisson Kernel on the Upper Half-Plane
For \(y > 0\) and \(x \in \mathbb{R}\), the
Poisson kernel on the upper half-plane is
\[
P_y(x) \;=\; \frac{1}{\pi} \cdot \frac{y}{x^2 + y^2}.
\]
Equivalently, \(P_y\) is the inverse Fourier transform of \(e^{-|\xi| y}\):
\(\widehat{P_y}(\xi) = e^{-|\xi| y}\).
The half-plane Poisson kernel is positive on its domain (\(y > 0\) makes both
numerator and denominator positive), and a direct integration gives
\[
\int_{-\infty}^{\infty} P_y(x)\, dx
\;=\; \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{y}{x^2 + y^2}\, dx
\;=\; \frac{1}{\pi}\left[\arctan\!\frac{x}{y}\right]_{-\infty}^{\infty}
\;=\; 1.
\]
With the kernel identified, the inverse transform expresses the solution as a
convolution.
Theorem: Poisson Integral Formula on the Half-Plane
Let \(f\) be a bounded continuous function on \(\mathbb{R}\). The function
\[
u(x, y)
\;=\; \int_{-\infty}^{\infty} P_y(x - s)\, f(s)\, ds
\qquad (y > 0)
\]
is harmonic on the upper half-plane \(\mathbb{H}\), extends continuously to
the closed half-plane \(\overline{\mathbb{H}}\) at every point of the
boundary, and satisfies \(u(x, 0) = f(x)\) on \(\partial\mathbb{H}\). It is
the unique solution of the Dirichlet problem on \(\mathbb{H}\) in the class
of bounded continuous extensions; uniqueness is established later as a
consequence of the maximum principle.
Proof:
Harmonicity: direct differentiation gives, for
\((x, y) \in \mathbb{H}\),
\[
\partial_{xx} \frac{y}{x^2 + y^2}
\;=\; \frac{2y(3x^2 - y^2)}{(x^2 + y^2)^3},
\qquad
\partial_{yy} \frac{y}{x^2 + y^2}
\;=\; \frac{2y(y^2 - 3x^2)}{(x^2 + y^2)^3},
\]
whose sum vanishes; hence \(\Delta P_y(x) = 0\) on \(\mathbb{H}\) as a
function of the two variables \((x, y)\). On each compact
\(K \subset \mathbb{H}\), the kernel and its second \((x,y)\)-derivatives
are bounded uniformly in \(s\) (since they decay like \(|x-s|^{-3}\) as
\(|s| \to \infty\) with \(y\) bounded away from \(0\) on \(K\)), so for
bounded \(f\) differentiation under the integral sign is valid on \(K\),
giving \(\Delta u(x, y) = \int \Delta_{(x,y)} P_y(x - s)\, f(s)\, ds = 0\)
on \(\mathbb{H}\).
Boundary behaviour: as \(y \to 0^+\), the kernel \(P_y(x)\)
concentrates near \(x = 0\) — its peak value at \(x = 0\) is
\(\frac{1}{\pi y}\), tending to \(+\infty\), while for any fixed \(x \neq 0\)
the denominator \(x^2 + y^2\) stays bounded below by \(x^2\) and
\(P_y(x) \to 0\). Combined with positivity and the normalization
\(\int_{-\infty}^\infty P_y(x)\, dx = 1\), this makes \(\{P_y\}_{y > 0}\) an
approximate identity on \(\mathbb{R}\) as \(y \to 0^+\).
For bounded continuous \(f\), fix \(x_0 \in \mathbb{R}\) and
\(\varepsilon > 0\). By continuity of \(f\) at \(x_0\), choose
\(\delta > 0\) so that \(|f(s) - f(x_0)| < \varepsilon\) whenever
\(|s - x_0| < \delta\). Using \(\int_{-\infty}^\infty P_y(x_0 - s)\, ds = 1\),
\[
u(x_0, y) - f(x_0)
\;=\; \int_{-\infty}^{\infty} P_y(x_0 - s)\,\bigl[f(s) - f(x_0)\bigr]\, ds,
\]
and splitting the integration domain into \(|s - x_0| < \delta\) and
\(|s - x_0| \geq \delta\):
\[
\bigl|u(x_0, y) - f(x_0)\bigr|
\;\leq\;
\varepsilon \int_{|s - x_0| < \delta} P_y(x_0 - s)\, ds
+ 2\|f\|_\infty \int_{|s - x_0| \geq \delta} P_y(x_0 - s)\, ds.
\]
The first integral is bounded by \(\int P_y = 1\), giving an
\(\varepsilon\) contribution. For the second, the change of variable
\(\sigma = s - x_0\) together with the symmetry \(P_y(-\sigma) = P_y(\sigma)\)
gives
\[
\int_{|s - x_0| \geq \delta} P_y(x_0 - s)\, ds
\;=\; \frac{2}{\pi}\int_\delta^{\infty} \frac{y}{\sigma^2 + y^2}\, d\sigma
\;=\; \frac{2}{\pi}\,\arctan\!\frac{y}{\delta},
\]
which tends to \(0\) as \(y \to 0^+\) (since \(\arctan(y/\delta) \to
\arctan 0 = 0\)). Choosing \(y\) small enough makes the second contribution
less than \(\varepsilon\); hence \(|u(x_0, y) - f(x_0)| \to 0\) as
\(y \to 0^+\). Since \(x_0 \in \mathbb{R}\) was arbitrary, \(u\) extends
continuously to \(\overline{\mathbb{H}}\) at every boundary point with the
boundary value \(u(x_0, 0) = f(x_0)\).
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 a kernel parametrized by the time
variable \(t\) — the heat kernel
\[
K_t(x) = (4\pi k t)^{-1/2} e^{-x^2/(4kt)},
\]
which concentrates onto the initial datum as \(t \to 0^+\) and provides the
solution by convolution. The wave equation admitted no such kernel
representation: d'Alembert's formula is a different structural object, not a
convolution with a single approximate identity. The Laplace equation on the
disk and on the half-plane both admit kernel representations — Poisson kernels
— 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 two specific domains
— the disk and the half-plane — and to the coordinate systems adapted to those
domains. We now turn to two qualitative properties that hold for
every harmonic function on any open subset of the plane, irrespective
of domain shape: the mean value property and the
maximum principle. Both will be derived from the Poisson
integral on the disk, but their conclusions are domain-independent and supply
the analytic structure that, in the heat-and-wave parallel, was carried by
smoothing and energy conservation respectively. The maximum principle in
particular is the signature property of the elliptic case — the pointwise
bound on the solution by its boundary data — and is the result that discharges
the marker placed in
the heat-equation treatment,
where the boundary-data form of the principle was promised but not derived.
Definition: Harmonic Function
Let \(\Omega \subset \mathbb{R}^2\) be open. A function \(u : \Omega \to
\mathbb{R}\) of class \(C^2\) is called harmonic on
\(\Omega\) if
\[
\Delta u(x, y) \;=\; \partial_{xx} u(x, y) + \partial_{yy} u(x, y) \;=\; 0
\qquad \text{for every } (x, y) \in \Omega.
\]
The Mean Value Property
The Poisson integral formula on the unit disk suggests the statement
of the mean value property already. 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 centre, to the plain average of its
boundary values. This special-case identity furnishes the form of the
mean value property; however, asserting it for an arbitrary harmonic
function on an arbitrary domain — not 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 (which is established later in this section as a consequence of
the maximum principle).
Theorem: Mean Value Property of Harmonic Functions
Let \(u\) be harmonic on an open set \(\Omega \subset \mathbb{R}^2\), and
let \(\overline{D(x_0, \rho)} \subset \Omega\) be a closed disk centred at
\(x_0\) with radius \(\rho > 0\). Then \(u(x_0)\) equals both the average
of \(u\) over the boundary circle and the average of \(u\) over the disk:
\[
u(x_0)
\;=\; \frac{1}{2\pi\rho}\int_{\partial D(x_0, \rho)} u\, ds
\;=\; \frac{1}{\pi\rho^2}\int_{D(x_0, \rho)} u\, dA,
\]
where \(ds\) is the arc-length element on the circle and \(dA\) the area
element on the disk.
Proof:
Boundary-average form: by translation it suffices to take
\(x_0 = 0\). Introduce polar coordinates \((r, \theta)\) centred at
the origin and write \(\tilde u(r, \theta) := u(r\cos\theta,
r\sin\theta)\) for the polar representation of \(u\), which is
\(C^2\) on the strip \((0, \rho_*) \times \mathbb{R}\) and
\(2\pi\)-periodic in \(\theta\) for each fixed \(r\), where
\(\rho_* > \rho\) is chosen so that
\(\overline{D(0, \rho_*)} \subset \Omega\). For
\(r \in [0, \rho_*)\) define the angular average
\[
I(r) \;=\; \frac{1}{2\pi}\int_0^{2\pi} \tilde u(r, \theta)\, d\theta.
\]
We will show \(I(r) = u(0)\) for every \(r \in [0, \rho_*)\);
applying this at \(r = \rho\) and using \(ds = \rho\, d\theta\) on
\(\partial D(0, \rho)\) will give the stated boundary-average
formula.
On each compact set \([r_1, r_2] \times [0, 2\pi]\) with
\(0 < r_1 < r_2 < \rho_*\), the function \(\tilde u\) and its
first- and second-order partial derivatives in \((r, \theta)\) are
continuous and hence bounded. Differentiation under the integral
sign is therefore valid on \((0, \rho_*)\), and
\[
I'(r)
\;=\; \frac{1}{2\pi}\int_0^{2\pi} \partial_r \tilde u(r, \theta)\, d\theta.
\]
Multiplying by \(r\) and differentiating once more,
\[
\begin{align*}
\bigl(r\, I'(r)\bigr)'
&=\; \frac{1}{2\pi}\int_0^{2\pi} \bigl(\partial_r \tilde u + r\, \partial_{rr} \tilde u\bigr)(r, \theta)\, d\theta \\\\
&=\; \frac{r}{2\pi}\int_0^{2\pi} \left(\partial_{rr} \tilde u + \frac{1}{r}\partial_r \tilde u\right)(r, \theta)\, d\theta.
\end{align*}
\]
The polar form of the Laplacian
\(\Delta u = \partial_{rr} \tilde u + \frac{1}{r}\partial_r \tilde u
+ \frac{1}{r^2}\partial_{\theta\theta} \tilde u\), derived earlier, and
the harmonicity assumption \(\Delta u = 0\) give
\(\partial_{rr} \tilde u + \frac{1}{r}\partial_r \tilde u
= -\frac{1}{r^2}\partial_{\theta\theta} \tilde u\) on
\((0, \rho_*) \times \mathbb{R}\). Substituting,
\[
\begin{align*}
\bigl(r\, I'(r)\bigr)'
&=\; -\frac{1}{2\pi r}\int_0^{2\pi} \partial_{\theta\theta} \tilde u(r, \theta)\, d\theta \\\\
&=\; -\frac{1}{2\pi r}\bigl[\partial_\theta \tilde u(r, \theta)\bigr]_{\theta=0}^{\theta=2\pi} \\\\
&=\; 0,
\end{align*}
\]
the last equality holding because \(\partial_\theta \tilde u\) is
\(2\pi\)-periodic in \(\theta\) (a consequence of the
\(2\pi\)-periodicity of \(\tilde u\) itself together with the
\(C^2\) regularity).
Hence \(r\, I'(r)\) is constant on \((0, \rho_*)\). Since
\(\partial_r \tilde u\) is continuous on the compact set
\(\overline{D(0, \rho_*/2)}\) and therefore bounded there, we have
\(|r\, I'(r)| \leq r \cdot \sup |\partial_r \tilde u| \to 0\) as
\(r \to 0^+\); the constant is therefore zero, and
\(I'(r) \equiv 0\) on \((0, \rho_*)\). The function \(I\) is also
continuous at \(r = 0\): the integrand
\(\tilde u(r, \theta) = u(r\cos\theta, r\sin\theta)\) is continuous in
\((r, \theta)\), so \(\lim_{r \to 0^+} \tilde u(r, \theta) = u(0)\)
uniformly in \(\theta \in [0, 2\pi]\), giving
\(\lim_{r \to 0^+} I(r) = u(0) = I(0)\). Combined with
\(I'(r) \equiv 0\) on \((0, \rho_*)\), this yields
\(I(r) = u(0)\) for every \(r \in [0, \rho_*)\). Evaluating at
\(r = \rho\) and using \(ds = \rho\, d\theta\) on
\(\partial D(0, \rho)\),
\[
\begin{align*}
u(0)
&=\; \frac{1}{2\pi}\int_0^{2\pi} u(\rho\cos\theta, \rho\sin\theta)\, d\theta \\\\
&=\; \frac{1}{2\pi\rho}\int_{\partial D(0, \rho)} u\, ds.
\end{align*}
\]
Reversing the translation reinstates a general centre \(x_0\).
Volume-average form: the boundary-average identity reads
\(\int_{\partial D(x_0, t)} u\, ds = 2\pi t\, u(x_0)\) for every
\(t \in (0, \rho]\). Integrating in \(t\) from \(0\) to \(\rho\),
\[
\begin{align*}
\int_0^\rho \int_{\partial D(x_0, t)} u\, ds\, dt
&=\; 2\pi\, u(x_0) \int_0^\rho t\, dt \\\\
&=\; \pi \rho^2\, u(x_0).
\end{align*}
\]
On the disk \(D(x_0, \rho)\), polar coordinates centred at \(x_0\)
give the area element \(dA = t\, dt\, d\theta\), and the
arc-length element on \(\partial D(x_0, t)\) is \(ds = t\, d\theta\);
hence
\[
\begin{align*}
\int_0^\rho \int_{\partial D(x_0, t)} u\, ds\, dt
&= \int_0^\rho \int_0^{2\pi} u(x_0 + t e^{i\theta})\, t\, d\theta\, dt \\\\
&= \int_{D(x_0, \rho)} u\, dA.
\end{align*}
\]
Equating and dividing by \(\pi \rho^2\) yields the volume-average
form.
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.
Theorem: Maximum Principle for Harmonic Functions
Let \(\Omega \subset \mathbb{R}^2\) be a bounded open set and let
\(u : \overline{\Omega} \to \mathbb{R}\) be continuous on
\(\overline{\Omega}\) and harmonic on \(\Omega\). Then \(u\) attains its
maximum and minimum on the boundary \(\partial\Omega\):
\[
\max_{\overline{\Omega}} u \;=\; \max_{\partial\Omega} u,
\qquad
\min_{\overline{\Omega}} u \;=\; \min_{\partial\Omega} u.
\]
In particular, for every interior point \(x \in \Omega\),
\[
\min_{\partial\Omega} u \;\leq\; u(x) \;\leq\; \max_{\partial\Omega} u.
\]
Proof:
The boundedness of \(\Omega\) makes \(\overline{\Omega}\) compact; the
continuity of \(u\) on \(\overline{\Omega}\) then guarantees that \(u\)
attains its maximum somewhere on \(\overline{\Omega}\). We show that the
maximum is attained on the boundary; the minimum claim follows by
applying the same argument to \(-u\), which is also harmonic.
Let \(M = \max_{\overline{\Omega}} u\) and define the set of maximum points
in the interior,
\[
E \;=\; \{x \in \Omega : u(x) = M\}.
\]
If \(E\) is empty, the maximum is attained only on the boundary and the
claim holds. Otherwise, we show that the connected component of \(\Omega\)
containing any \(x_0 \in E\) is contained entirely in \(E\): this forces
\(u \equiv M\) on that component and, by continuity on
\(\overline{\Omega}\), \(u \equiv M\) on its closure — in particular on
the boundary of the component, which lies in \(\partial\Omega\). The
maximum is therefore attained on \(\partial\Omega\) in this case as well.
Fix \(x_0 \in E\). Since \(\Omega\) is open, choose \(\rho > 0\) small
enough that \(\overline{D(x_0, \rho)} \subset \Omega\). The mean value
property gives
\[
M \;=\; u(x_0) \;=\; \frac{1}{\pi \rho^2}\int_{D(x_0, \rho)} u\, dA.
\]
Since \(u \leq M\) everywhere on \(\overline{\Omega}\), the integrand
\(M - u\) is non-negative on \(D(x_0, \rho)\). The identity above is
equivalent to
\[
\int_{D(x_0, \rho)} (M - u)\, dA = 0
\]
for a non-negative continuous integrand, a zero integral forces the integrand to vanish
identically (if \(M - u\) were positive at some point, continuity would
keep it positive on a neighbourhood of positive area, making the integral
strictly positive). Hence \(u \equiv M\) on \(D(x_0, \rho)\), and
\(D(x_0, \rho) \subset E\). This shows \(E\) is open as a subset of \(\Omega\).
The set \(E\) is also closed in \(\Omega\): if \(x_n \in E\) with
\(x_n \to x \in \Omega\), continuity of \(u\) gives \(u(x) = \lim u(x_n) =
M\), so \(x \in E\). Within the connected component of \(\Omega\)
containing \(x_0\), the set \(E\) is therefore both open and closed and
nonempty; by connectedness, \(E\) coincides with the entire component.
Thus \(u \equiv M\) on this component, and by continuity on
\(\overline{\Omega}\), \(u \equiv M\) on the closure of the component as
well — in particular on its boundary, which is part of \(\partial\Omega\).
The maximum \(M\) is therefore attained on \(\partial\Omega\).
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 formulas
constructed earlier therefore give the solution of the Dirichlet
problem on the disk and on the half-plane respectively, not merely a
solution. This is the elliptic analogue of the energy-based uniqueness
established for the wave equation, with the controlling object replaced from
a global quadratic invariant to a bound by the extremes of the boundary data.
This discharges the marker placed in the introduction: the boundary-data
form of the principle, promised but not yet derived, is now established.
The structural placement of the elliptic case alongside the parabolic
pointwise bound and the hyperbolic energy invariant — three controlling
quantities, two pointwise sup/inf bounds and one integral conserved energy —
is codified in the closing section.
The Algebraic Origin of the Trichotomy
The three classical PDEs treated on this site 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 behaviour at each point
is governed by the discriminant \(B^2 - 4AC\) — exactly the same expression
that classifies plane conic sections \(A x^2 + B xy + C y^2 + \ldots = 0\).
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 \(B^2 - 4AC = -4 \det M\): the discriminant records the sign of the
determinant of \(M\), which in turn records the sign pattern of its
eigenvalues. Negative discriminant means \(\det M > 0\), so the two
eigenvalues of \(M\) share the same sign — the matrix is definite, and the
PDE is elliptic; positive discriminant means \(\det M < 0\), so the
eigenvalues have opposite signs — the matrix is indefinite, and the PDE is
hyperbolic; zero discriminant means \(\det M = 0\), so one eigenvalue
vanishes — the matrix is 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 \((-k, 0)\). The shared names
— elliptic, parabolic, hyperbolic — are not metaphor: they record the
eigenvalue sign pattern of a \(2 \times 2\) symmetric matrix, and the same
classification extends verbatim to higher-dimensional PDEs, where the
principal symbol is a larger symmetric matrix and the discriminant
\(B^2 - 4AC\) gives way to a richer signature.
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,
identifying the second variable \(y\) with time \(t\), the principal part
\[
\partial_{tt} u - c^2 \partial_{xx} u
\]
gives
\[
A = -c^2, \, B = 0, \, C = 1
\]
and the characteristic equation
\[
-c^2(dt/dx)^2 + 1 = 0
\]
has the two real roots
\[
dt/dx = \pm 1/c, \text{ or equivalently } dx/dt = \pm c.
\]
These define two real characteristic families — the lines
\[
x \pm ct = \text{const}
\]
along which d'Alembert's formula carried the travelling waves of the wave page.
For the heat equation, with principal part \(-\partial_{xx} u\)
(the first-order term \(\partial_t u\) does not enter the classification) and
degenerate quadratic in \(dt/dx\), 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
— 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 discontinuous boundary datum for the wave equation persists
as a discontinuity travelling along a characteristic line; for the Laplace
equation, harmonic functions are smooth — even real-analytic — at every
interior point regardless of boundary regularity. The maximum-principle
machinery developed earlier supplies the qualitative side of this statement
(pointwise boundedness by boundary data); the full interior-analyticity
result belongs to elliptic regularity theory and is not developed here.
The three controlling quantities catalogued in the maximum-principle
section — pointwise sup/inf bound by initial data for heat, integral energy
invariant for wave, pointwise min/max bound by boundary data for Laplace —
are three faces of this single algebraic fact: the eigenvalue sign pattern
of the principal-part matrix. Two-characteristic propagation makes the wave equation conserve
mechanical energy along its trajectories; 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, propagation — and each admits a discrete dual
treated elsewhere on this site: the
graph Laplacian
as the discrete Dirichlet energy operator, and the factorization
\(\boldsymbol{L} = \boldsymbol{B}\boldsymbol{B}^\top\) in terms of the
incidence matrix
of a graph as the discrete counterpart of \(-\Delta\), the positive-definite version of the continuous Laplacian.
The parallel running through the trichotomy is summarized below.
|
Elliptic (Laplace) |
Parabolic (Heat) |
Hyperbolic (Wave) |
| Equation |
\(\partial_{xx} u + \partial_{yy} u = 0\) |
\(\partial_t u - k\, \partial_{xx} u = 0\) |
\(\partial_{tt} u - c^2 \partial_{xx} u = 0\) |
| Matrix \(M\) |
\(\operatorname{diag}(1,\, 1)\) |
\(\operatorname{diag}(-k,\, 0)\) |
\(\operatorname{diag}(-c^2,\, 1)\) |
| Eigenvalue signs |
\((+,\, +)\) — definite |
\((-,\, 0)\) — degenerate |
\((-,\, +)\) — indefinite |
| Discriminant \(B^2 - 4AC\) |
\(-4 < 0\) |
\(0\) |
\(4c^2 > 0\) |
| Real characteristics |
none |
one family: \(t = \text{const}\) |
two families: \(x \pm ct = \text{const}\) |
| Controls solution |
boundary data |
initial data |
initial data (energy conserved) |
| Bound by data extremes |
\(\min_{\partial\Omega} u \leq u \leq \max_{\partial\Omega} u\) |
\(\inf f \leq u \leq \sup f\) |
generally none — integral invariant \(E(t) = E(0)\) |
| Interior regularity |
analytic |
\(C^\infty\) (analytic in \(x\)) |
singularities persist along characteristics |
| Paradigm |
harmonic interpolation |
dissipation |
propagation |
One algebraic fact — the eigenvalue sign pattern of a \(2 \times 2\) symmetric matrix — produces, across its three
possible patterns \((+,+)\) (definite, elliptic), \((-,+)\) (indefinite, hyperbolic), \((-,0)\) (degenerate, parabolic),
the analytic, geometric, and discrete structure of the entire trichotomy.
Interactive Demonstration
The widget below acts on the same initial bump with all three equations side by side: heat diffuses it forward in time,
wave splits it into travelling components, and Laplace propagates it inward from the boundary along increasing depth \(y\).
The three controlling quantities just catalogued — initial-data bound for heat, energy invariant for wave,
boundary-data bound for Laplace — appear directly as readouts beneath each panel.
A note on the wave panel's energy readout.
When parts of the travelling wave reach the window's edge — most easily with the bimodal profile, whose outermost peaks
begin at \(x = \pm 0.4\), but also at large wave speeds with the centred profiles — the displayed energy ratio
\(E(t)/E(0)\) drops visibly below unity and is flagged in amber or red. This is not a failure of energy conservation
for the wave equation — the conservation theorem on the unbounded line is exact — but a finite-window artifact:
waves that leave the visible interval cease to contribute to the integral
\[
\tfrac{1}{2}\int_{-1}^{1}\!\bigl[(\partial_t u)^2 + c^2(\partial_x u)^2\bigr] dx
\]
actually being computed. The same universal problem haunts every numerical simulation of wave propagation on a finite
grid, and is the reason absorbing boundary conditions (Mur conditions, perfectly matched layers, sponge regions)
were invented.
The role of the initial velocity g.
The wave equation is second order in time, so a complete initial datum needs both the initial shape \(u(\cdot, 0) = f\) and
the initial velocity \(u_t(\cdot, 0) = g\) — distinguishing it sharply from the parabolic heat equation (one initial datum)
and the elliptic Laplace equation (a stationary problem with no time variable). With \(g \equiv 0\)
(the demo's "at rest" preset), the d'Alembert formula reduces to the arithmetic mean
\[
u(x, t) = \tfrac{1}{2}\bigl[\tilde f(x - ct) + \tilde f(x + ct)\bigr],
\]
so the solution is automatically bounded between \(\min f\) and \(\max f\) — a degenerate consequence of
the vanishing initial velocity, not a property of the wave equation itself. Switching the "Wave initial velocity"
selector to outward or uniform turns on the integral term
\[
\tfrac{1}{2c}\!\int_{x - ct}^{x + ct} \tilde g(s)\, ds,
\]
which can carry the solution above \(\max f\) or below \(\min f\). The \(\max u\) readout below the wave panel
will then exceed \(1\) and turn amber — the visual confirmation that, unlike the heat and Laplace equations,
the wave equation admits no general pointwise bound by its data, only an integral (energy) invariant. The dotted
teal curve in the wave panel, identified in the legend as \(g(x)\), shows the second initial datum that hyperbolic
dynamics requires.