Classification of Second-Order Linear PDEs

Nearly every field equation of classical physics is a second-order linear partial differential equation, and a first glance flatters them with similarity: each is a sum of second derivatives of one field set equal to a source. Yet the wave, heat, and Laplace equations behave in ways that could hardly be more different. A wave carries a disturbance across a room at a fixed speed and keeps its sharp edges; heat blurs a hot spot into featurelessness and forgets how it started; a static potential holds a rigid tension in which the value at the centre is fixed entirely by what happens on the walls. This trichotomy — propagation, diffusion, equilibrium — is no accident of the examples. It is forced by a single piece of algebra applied to the coefficients of the highest derivatives, the principal symbol, and nothing else. The organising quantity is the discriminant $B^2-AC$ of that principal symbol, exactly the invariant that sorts the conic sections into ellipse, parabola, and hyperbola — the origin of the three type-names. Its sign decides whether the operator has no real characteristics (elliptic), one repeated family (parabolic), or two distinct families (hyperbolic), and that in turn decides which auxiliary data make the problem well-posed in the sense of Hadamard. Get the type right and the whole qualitative story — smoothing versus signalling, boundary versus initial data, maximum principles versus domains of dependence — follows before a single solution is computed. This article builds the classification from the quadratic form, proves it coordinate-independent, and shows why matching the wrong data to the wrong type is a catastrophe.

Key takeaways

A linear second-order PDE in two variables, $A u_{xx}+2B u_{xy}+C u_{yy}+\dots=0$ , is classified entirely by the sign of the discriminant $B^2-AC$ of its principal symbol — the highest-derivative coefficients alone.
Three types result: elliptic ( $B^2-AC<0$ , prototype Laplace $\nabla^2u=0$ ), parabolic ( $B^2-AC=0$ , prototype heat $u_t=\kappa u_{xx}$ ), and hyperbolic ( $B^2-AC>0$ , prototype wave $u_{tt}=c^2u_{xx}$ ).
The type is the definiteness of the quadratic form $A\xi^2+2B\xi\eta+C\eta^2$ — the same invariant $B^2-AC$ that classifies conic sections, unchanged by any nonsingular coordinate change.
Each type demands its own well-posed data: elliptic ↔ boundary values on a closed surface, parabolic and hyperbolic ↔ initial data plus boundary conditions; mismatching them is Hadamard's ill-posed problem.
Characteristics — curves along which the highest derivatives are undetermined — are real and distinct (hyperbolic), real and coincident (parabolic), or complex (elliptic), and set the finite propagation speed of a wave.
In $n$ variables the discriminant becomes the eigenvalue signature of the coefficient matrix $a^{ij}$ ; the wave operator's signature $(n-1,1)$ separates it from the Laplacian's $(n,0)$ .
Key idea: the type of a second-order linear PDE is fixed by its principal symbol alone; the sign of $B^2-AC$ — equivalently the eigenvalue signature of the coefficient matrix — sorts it into elliptic, parabolic, or hyperbolic, and that choice fixes which data make the problem well-posed.

The core insight

The type of a second-order linear PDE depends only on its principal symbol; the sign of the discriminant $B^2-AC$ (equivalently the eigenvalue signature of the coefficient matrix) classifies it as elliptic, parabolic, or hyperbolic, and this fixes the well-posed data.

Where this lives today

The trichotomy is the first decision in any computational physics pipeline, because it dictates the numerical method as sharply as the data. Computational fluid dynamics lives on the elliptic–hyperbolic boundary: a transonic wing carries a subsonic (elliptic) region and a supersonic (hyperbolic) pocket split by a sonic line — exactly the Tricomi picture — and solvers must switch discretisation, central differencing where the flow is elliptic and upwind schemes where it is hyperbolic, or the captured shocks are nonsense. In numerical relativity the Einstein equations become a hyperbolic evolution off an elliptic constraint surface, and choosing a strongly hyperbolic formulation is what separates a stable binary-black-hole merger simulation — the kind now underpinning LIGO–Virgo waveform templates — from one that blows up in milliseconds. The same taxonomy governs finite elements for elliptic problems, implicit time-stepping for parabolic ones, and the CFL stability condition — a descendant of the hyperbolic domain of dependence — that caps the time step of every explicit wave solver. It remains a live frontier too: the well-posedness of the mixed-type Euler and Navier–Stokes systems, whose type shifts with the flow, sits among the hardest open problems in analysis.

The physical picture

Attach to each point the symmetric matrix of second-derivative coefficients and diagonalise it. Locally the operator becomes $\lambda_1\partial_1^2+\lambda_2\partial_2^2+\dots$ , and the whole story lives in the signs of the eigenvalues. If every $\lambda$ shares one sign the operator is a stretched Laplacian — isotropic and smoothing, its solutions as rigid as a soap film — the elliptic case. If exactly one $\lambda$ flips sign, that direction behaves like time and the operator is a stretched d'Alembertian $\partial_t^2-c^2\nabla^2$ : disturbances propagate, and the equation is hyperbolic. If a $\lambda$ vanishes the form degenerates and the equation is parabolic. The first figure draws the conic each principal symbol carves out.

Figure

The discriminant Δ = B² − AC sorts the principal symbol A ξ² + 2B ξ η + C η² into three conics: Δ negative gives an ellipse and an elliptic PDE (∇²u = 0, eigenvalue signs ++), Δ zero a parabola and a parabolic PDE (uₜ = κ uₓₓ, one eigenvalue zero), Δ positive a hyperbola and a hyperbolic PDE (uₜₜ = c² uₓₓ, signs + −)

The elliptic–hyperbolic divide is really about how information travels. Elliptic equations have only complex characteristics, so no real surface shields a point from the rest of the domain — perturb the boundary anywhere and every interior value responds, so a potential is rigid and obeys a maximum principle. Hyperbolic equations have two real characteristic families along which the highest derivatives are undetermined, so discontinuities ride them at a finite speed $c$ and a point at time $t$ feels only the initial data in a bounded interval, its domain of dependence. The second figure draws that domain for the wave equation and turns "finite propagation speed" into a measured length.

Figure

Domain of dependence for the wave equation uₜₜ = c² uₓₓ with speed c = 2: the two characteristics x − ct = const and x + ct = const through the point P at x₀ = 4, t₀ = 2 strike the initial line t = 0 at x = x₀ − ct₀ = 0 and x = x₀ + ct₀ = 8, so the solution at P depends only on the initial data in the interval 0 to 8

The core equations

Everything begins with the most general linear second-order equation in two independent variables. Only the first three coefficients — those of the second derivatives — will end up mattering for the type; the rest are along for the ride.

A\,u_{xx}+2B\,u_{xy}+C\,u_{yy}+D\,u_x+E\,u_y+F\,u=G \qquad (1)

where: $u(x,y)$ — the unknown field; $A,B,C$ — principal coefficients, possibly functions of $x,y$ ; $D,E,F$ — lower-order coefficients; $G$ — the source. Reading it: only the leading form in $A,B,C$ decides the type, and the factor $2B$ makes the symbol matrix exactly $\begin{pmatrix}A&B\\B&C\end{pmatrix}$ . The classifier is the discriminant of that leading part — a single number whose sign is the entire taxonomy.

\Delta = B^2 - AC \qquad (2)

where: $\Delta$ — the discriminant: $\Delta<0$ elliptic, $\Delta=0$ parabolic, $\Delta>0$ hyperbolic. Reading it: this is $-\det\begin{pmatrix}A&B\\B&C\end{pmatrix}=-\lambda_1\lambda_2$ , so equal-sign eigenvalues give $\Delta<0$ , opposite signs $\Delta>0$ , a zero eigenvalue $\Delta=0$ . The characteristics are the curves along which the second derivatives cannot be recovered from Cauchy data. Their slopes obey a quadratic whose discriminant is again $\Delta$ , which is why the same number governs both the algebra and the geometry.

A\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2-2B\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)+C=0 \qquad (3)

where: $\mathrm{d}y/\mathrm{d}x$ — the slope of a characteristic curve $\phi(x,y)=\text{const}$ . Reading it: a curve is characteristic exactly when the principal part degenerates along it — this is that condition written for the slope. Solving the quadratic exposes the trichotomy directly in the number of real characteristic directions.

\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{B\pm\sqrt{B^2-AC}}{A} \qquad (4)

where: the two roots are distinct real slopes when $\Delta>0$ (hyperbolic), one repeated real slope when $\Delta=0$ (parabolic), and a complex-conjugate pair when $\Delta<0$ (elliptic). Reading it: real characteristics are the tracks signals travel along, so their count is the physical difference between the regimes. Adopting the characteristics as new coordinates strips each equation down to a canonical form in which the type is manifest and the lower-order terms are all that remain to distinguish one member from another.

\underbrace{u_{\xi\eta}=\dots}_{\text{hyperbolic}}\qquad \underbrace{u_{\xi\xi}+u_{\eta\eta}=\dots}_{\text{elliptic}}\qquad \underbrace{u_{\eta\eta}=\dots}_{\text{parabolic}} \qquad (5)

where: $\xi,\eta$ — canonical coordinates from the characteristics (both real for hyperbolic; real and imaginary parts of a complex pair for elliptic; one characteristic plus a transverse coordinate for parabolic). Reading it: after this reduction every hyperbolic equation is a wave equation in disguise, every elliptic one a Laplace equation, every parabolic one a heat equation. In more than two variables the scalar discriminant has no meaning, but the principle survives: the symmetric coefficient matrix has real eigenvalues, and their signs — the signature — are the invariant.

\sum_{i,j=1}^{n} a^{ij}\,\partial_i\partial_j u + \text{(lower order)} = 0,\qquad a^{ij}\to\operatorname{diag}(\lambda_1,\dots,\lambda_n) \qquad (6)

where: $a^{ij}=a^{ji}$ — the symmetric principal-coefficient matrix; $\lambda_k$ — its eigenvalues. Reading it: elliptic means all $\lambda_k$ of one sign; hyperbolic means exactly one opposite sign, signature $(n-1,1)$ , as in the wave operator; parabolic means one $\lambda_k=0$ with the rest of one sign; more of each sign is the rarer ultrahyperbolic case.

Derivation AReducing a hyperbolic equation to its canonical form

The goal is to take any hyperbolic equation $A u_{xx}+2B u_{xy}+C u_{yy}=0$ with $B^2-AC>0$ and, by adopting the characteristic curves as new coordinates $\xi=\phi(x,y)$ , $\eta=\psi(x,y)$ , transform it into the normal form $u_{\xi\eta}=\text{lower order}$ in which it is visibly a wave equation. The engine is the chain rule on the second derivatives, and the payoff is that the constant-coefficient case reproduces d'Alembert's coordinates $x\pm ct$ exactly.

Step 1. Under $\xi=\phi(x,y)$ , $\eta=\psi(x,y)$ the chain rule turns the operator into $\bar A\,u_{\xi\xi}+2\bar B\,u_{\xi\eta}+\bar C\,u_{\eta\eta}+(\text{lower order})$ , where the new leading coefficient is the same quadratic form on the new coordinate's gradient: $\bar A=A\phi_x^2+2B\phi_x\phi_y+C\phi_y^2$ , and $\bar C$ the identical expression in $\psi$ . Why this matters: the principal symbol transforms as a quadratic form, so the coordinate change cannot alter its intrinsic nature, only its appearance.

Step 2. Demand the pure second derivatives vanish, $\bar A=\bar C=0$ , i.e. $A\phi_x^2+2B\phi_x\phi_y+C\phi_y^2=0$ . Dividing by $\phi_y^2$ and writing $-\phi_x/\phi_y=\mathrm{d}y/\mathrm{d}x$ recovers the characteristic equation $A(\mathrm{d}y/\mathrm{d}x)^2-2B(\mathrm{d}y/\mathrm{d}x)+C=0$ . Why this matters: the coordinate curves that annihilate the pure second derivatives are exactly the characteristics — the reduction is a geometric alignment, not a trick.

Step 3. Because $B^2-AC>0$ , the characteristic equation has two distinct real roots $\mathrm{d}y/\mathrm{d}x=(B\pm\sqrt{B^2-AC})/A$ , giving two independent real families and hence two coordinates $\xi,\eta$ . Why this matters: distinct real slopes make $\xi$ and $\eta$ independent, so the change of variables is invertible — the parabolic case fails here because the roots coincide.

Step 4. With $\bar A=\bar C=0$ the surviving cross coefficient $\bar B=A\phi_x\psi_x+B(\phi_x\psi_y+\phi_y\psi_x)+C\phi_y\psi_y$ is nonzero for independent characteristics; dividing by $2\bar B$ leaves precisely $u_{\xi\eta}=\text{lower order}$ . Why this matters: the single mixed derivative is the fingerprint of two-way propagation, making the general solution a sum of a left-mover and a right-mover.

Step 5. For constant coefficients the coordinates integrate to straight lines $\xi=y-\frac{B-\sqrt{B^2-AC}}{A}x$ and $\eta=y-\frac{B+\sqrt{B^2-AC}}{A}x$ ; for the wave equation $u_{tt}-c^2u_{xx}=0$ ( $A=-c^2$ , $B=0$ , $C=1$ ) these collapse to $\xi=x-ct$ and $\eta=x+ct$ , and $u_{\xi\eta}=0$ integrates to d'Alembert's $u=f(x-ct)+g(x+ct)$ . Why this matters: the abstract reduction lands on the most familiar solution in physics — the canonical form is the travelling-wave decomposition seen from a distance. *The chain-rule reduction and the wave-equation specialization $\xi=x-ct,\ \eta=x+ct$ confirm the canonical form; the roots of $m^2-4m+3$ are $3$ and $1$ , as used in the worked example.*

Go deeper · Derivation B · 1 minThe discriminant is a coordinate-independent invariant

Derivation BThe discriminant is a coordinate-independent invariant

The goal is to prove that the type — the sign of $B^2-AC$ — cannot be changed by any nonsingular linear change of independent variables, so "elliptic", "parabolic", and "hyperbolic" are intrinsic labels, not artefacts of the chosen coordinates. The strategy: a linear coordinate change acts on the symbol matrix by congruence, which multiplies the determinant by a positive square and so preserves its sign.

Step 1. Under $\xi=\alpha x+\beta y$ , $\eta=\gamma x+\delta y$ with Jacobian $J=\alpha\delta-\beta\gamma\neq0$ , the symbol matrix transforms by congruence, $\bar M=M^{\mathsf T}\begin{pmatrix}A&B\\B&C\end{pmatrix}M$ with $M=\begin{pmatrix}\alpha&\gamma\\\beta&\delta\end{pmatrix}$ . Why this matters: a quadratic form pulled back through a linear map is exactly a congruence — the precise statement of rewriting the principal part in new coordinates.

Step 2. Determinants obey $\det(M^{\mathsf T}NM)=(\det M)^2\det N$ , so $\det\bar M=J^2\det\begin{pmatrix}A&B\\B&C\end{pmatrix}$ . Why this matters: the symbol determinant is $AC-B^2=-\Delta$ , so tracking it tracks the discriminant.

Step 3. Reading off the entries, $\bar A\bar C-\bar B^2=J^2(AC-B^2)$ , equivalently $\bar B^2-\bar A\bar C=J^2\,(B^2-AC)$ . Why this matters: the new discriminant is the old one scaled by the strictly positive $J^2$ , which changes its magnitude but never its sign.

Step 4. Since $J^2>0$ , the sign of $B^2-AC$ — and with it the type — is preserved. This is Sylvester's law of inertia for $2\times2$ forms: the signature, not the coefficients, is the invariant. Why this matters: classify an equation once in any frame and it is classified forever — the taxonomy is a property of the physics, not the chart. *The algebraic identity $\bar B^2-\bar A\bar C=J^2(B^2-AC)$ is the direct expression of Sylvester's law of inertia: the signature, not the individual coefficients, is the invariant.*

Go deeper · Derivation C · 1 minWhy Cauchy data ruins an elliptic equation — Hadamard's example

Derivation CWhy Cauchy data ruins an elliptic equation — Hadamard's example

The goal is to show, with an explicit family of solutions, that feeding an elliptic equation the Cauchy data appropriate to a hyperbolic one destroys well-posedness: the solution stops depending continuously on the data, so an arbitrarily small input change produces an arbitrarily large output. We build Hadamard's counterexample for the Laplace equation and watch the catastrophe unfold as a single integer is turned up.

Step 1. Take the elliptic equation $u_{xx}+u_{yy}=0$ on $y\ge 0$ with Cauchy data on $y=0$ : $u(x,0)=0$ and $u_y(x,0)=\tfrac1n\sin(nx)$ for a positive integer $n$ . The data is as small as we like, its size $1/n\to 0$ as $n\to\infty$ . Why this matters: Cauchy data is exactly what makes the hyperbolic wave equation well-posed, so imposing it here tests whether type really matters.

Step 2. The solution is $u(x,y)=\tfrac{1}{n^2}\sinh(ny)\sin(nx)$ : $u_{xx}=-\sinh(ny)\sin(nx)$ and $u_{yy}=+\sinh(ny)\sin(nx)$ sum to zero, $u(x,0)=0$ , and $u_y(x,0)=\tfrac1n\sin(nx)$ , matching the data exactly. Why this matters: it is a genuine, exact solution, so the blow-up below is no approximation artefact.

Step 3. Fix any $y>0$ and let $n\to\infty$ . The data amplitude $1/n$ shrinks to zero, but $\sinh(ny)\sim\tfrac12 e^{ny}$ , so the solution amplitude $\tfrac{1}{n^2}\sinh(ny)$ grows without bound — vanishingly small data produces unboundedly large solutions. Why this matters: this is the failure of continuous dependence, the third of Hadamard's conditions (existence, uniqueness, stability), caused solely by feeding an elliptic equation hyperbolic-style data.

Step 4. Contrast the correct pairing: with boundary data on a closed curve the same Laplace equation is well-posed and obeys a maximum principle, so small boundary changes give small interior changes. Why this matters: the trichotomy is the rule for which data to supply, and violating it makes an otherwise sensible equation physically useless. *Direct substitution confirms that $u=n^{-2}\sinh(ny)\sin(nx)$ solves $u_{xx}+u_{yy}=0$ with the stated data, and that $n^{-2}\sinh(ny)\to\infty$ at fixed $y>0$ while the data amplitude $n^{-1}\to 0$ — the Hadamard instability in explicit form.*

Limiting cases

The three archetypes are the three signs of $\Delta=B^2-AC$ ; the fourth row shows that when the coefficients vary in space the type itself can change from region to region.

Case	Result	Meaning
Elliptic	$B^2-AC<0\ \Rightarrow\ u_{xx}+u_{yy}=0$	No real characteristics; smooth, rigid interior tied to closed-boundary data, with a maximum principle. Prototype: Laplace.
Parabolic	$B^2-AC=0\ \Rightarrow\ u_t=\kappa\,u_{xx}$	One repeated real characteristic; infinite signal speed, instantaneous smoothing, irreversible in time. Prototype: heat.
Hyperbolic	$B^2-AC>0\ \Rightarrow\ u_{tt}=c^2u_{xx}$	Two real distinct characteristics $x\pm ct=\text{const}$ ; finite propagation speed, discontinuities preserved along them. Prototype: wave.
Variable type (Tricomi)	$y\,u_{xx}+u_{yy}=0,\ \Delta=-y$	$A=y,\ B=0,\ C=1$ : elliptic for $y>0$ , parabolic on the sonic line $y=0$ , hyperbolic for $y<0$ . The model of transonic flow.

Common trapOnly the top-order coefficients count — and the type can move

Two mistakes are perennial. The first is to think a first-derivative or source term can change the type: it cannot. The classification reads only the principal symbol, so $u_{xx}-4u_{xy}+4u_{yy}+u_x=0$ is parabolic ( $\Delta=0$ ) regardless of the $u_x$ , and adding a source $G$ or a mass term $Fu$ never shifts it. The second is to assume the type is a global label. When $A,B,C$ depend on position the discriminant is a function whose sign can flip across the domain: the Tricomi equation $y\,u_{xx}+u_{yy}=0$ is elliptic above $y=0$ and hyperbolic below, so the same equation smooths in one half-plane and propagates in the other. Always classify pointwise before deciding what data to impose.

Worked examples

1 · Classify an equation and reduce it to canonical form (fully worked)

Classify $u_{xx}+4u_{xy}+3u_{yy}=0$ , find its characteristics, and write down its general solution.

Step 1. Read off the coefficients ( $2B$ convention): $A=1$ , $2B=4\Rightarrow B=2$ , $C=3$ . The discriminant is $\Delta=B^2-AC=2^2-(1)(3)=\mathbf{1}$ . Since $\Delta>0$ the equation is hyperbolic — expect two real characteristic families and travelling-wave behaviour.

Step 2. Solve $m^2-4m+3=0$ with $m=\mathrm{d}y/\mathrm{d}x$ : the roots are $m=\dfrac{4\pm\sqrt{16-12}}{2}=\mathbf{3\ \text{and}\ 1}$ , two distinct real slopes confirming the verdict.

Step 3. Integrate: $\mathrm{d}y/\mathrm{d}x=3$ gives $y-3x=\text{const}$ and $\mathrm{d}y/\mathrm{d}x=1$ gives $y-x=\text{const}$ ; adopt these as canonical coordinates $\xi=y-3x$ , $\eta=y-x$ .

Step 4. The equation becomes the canonical form $u_{\xi\eta}=0$ (no lower-order terms survive). Integrating in $\eta$ then $\xi$ gives the general solution as a sum of two travelling profiles: $\boxed{u(x,y)=f(y-3x)+g(y-x)}$ , each constant along one characteristic family.

2 · A mixed-type equation: the Tricomi equation of transonic flow

Classify $y\,u_{xx}+u_{yy}=0$ throughout the plane and find its characteristics where they are real.

Step 1. Here $A=y$ , $B=0$ , $C=1$ , so $\Delta=B^2-AC=-y$ : elliptic for $y>0$ , parabolic on the sonic line $y=0$ , hyperbolic for $y<0$ . One equation carries all three types, split by a horizontal line — the skeleton of subsonic, sonic, and supersonic flow.

Step 2. Where it is hyperbolic ( $y<0$ ) the characteristic equation $y(\mathrm{d}y/\mathrm{d}x)^2+1=0$ gives $(\mathrm{d}y/\mathrm{d}x)^2=-1/y>0$ , so $\mathrm{d}y/\mathrm{d}x=\pm 1/\sqrt{-y}$ , two real slopes.

Step 3. Integrate $\sqrt{-y}\,\mathrm{d}y=\pm\,\mathrm{d}x$ to get $x\mp\tfrac{2}{3}(-y)^{3/2}=\text{const}$ , a pair of semicubical parabolas meeting the sonic line tangentially. As a check, at $y=-4$ one branch reaches $x-\tfrac{2}{3}(4)^{3/2}=x-\mathbf{5.33}$ . These are the Mach lines along which supersonic disturbances propagate; above $y=0$ the characteristics turn complex and the flow is elliptic and shock-free.

3 · Signature in four dimensions: wave operator versus Laplacian

Classify the 4-dimensional Laplacian $\partial_x^2+\partial_y^2+\partial_z^2+\partial_w^2$ and the d'Alembertian $\partial_t^2-c^2(\partial_x^2+\partial_y^2+\partial_z^2)$ by eigenvalue signature.

Step 1. The 4D Laplacian has matrix $\operatorname{diag}(1,1,1,1)$ , signature $(4,0)$ — all eigenvalues of one sign, so elliptic: smooth harmonic solutions with no propagation.

Step 2. The d'Alembertian has matrix $\operatorname{diag}(1,-c^2,-c^2,-c^2)$ , signature $(1,3)$ — one sign isolated from the rest, the $(n-1,1)$ pattern, so hyperbolic, with the time direction special and a light-cone domain of dependence. The lone sign flip is the entire difference between static equilibrium and relativistic propagation.

Physics of the portrait

The classification of PDEs is, at bottom, a theory of what data a physical law needs to have a unique, stable answer — the achievement of Jacques Hadamard, who turned "the right boundary conditions" from folklore into a precise mathematical requirement.

Origins · 1 minJacques Hadamard and how this was discovered

Jacques Hadamard

1865–1963 · well-posedness and the classification of partial differential equations

Jacques Hadamard was one of the last universalists of French mathematics, ranging over number theory, complex analysis, geometry, and physics; physicists remember him chiefly for making the link between a PDE's type and its well-posedness rigorous. The naming came earlier — Paul du Bois-Reymond christened equations "elliptic", "parabolic", and "hyperbolic" in 1889 by analogy with the conics their principal symbols trace out. But it was Hadamard, in his 1902 essay on boundary-value problems and his 1923 lectures on the Cauchy problem, who laid down the three conditions a physically sensible problem must satisfy: a solution must exist, be unique, and depend continuously on the data. His deliberately shocking counterexample — small Cauchy data driving the Laplace equation to unbounded solutions — showed that matching the wrong data to an elliptic equation is not a technicality but a failure of physics, and made "well-posed in the sense of Hadamard" the standard against which every field equation is now measured.

History

1747

— d'Alembert gives the travelling-wave solution of the vibrating string, the first hyperbolic prototype

1822

— Fourier's analytic theory of heat establishes the parabolic diffusion prototype

1889

— du Bois-Reymond names the three classes elliptic, parabolic, and hyperbolic after the conics

1902

— Hadamard defines well-posedness and exhibits an ill-posed Cauchy problem for the Laplace equation

1923

— Tricomi analyses the mixed elliptic–hyperbolic equation modelling transonic flow

Going further

Texts. Stone & Goldbart, Mathematics for Physics, Ch. 6 (Partial Differential Equations) · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 9 · Riley, Hobson & Bence, Mathematical Methods for Physics and Engineering, 3rd ed., Ch. 20 · Courant & Hilbert, Methods of Mathematical Physics, Vol. II, Ch. III · Evans, Partial Differential Equations, 2nd ed., §1.2. The key checkable results in this article are: the discriminant signs for Laplace ( $\Delta<0$ ), heat ( $\Delta=0$ ), and wave ( $\Delta>0$ ); the chain-rule reduction to $u_{\xi\eta}=0$ with wave specialization $\xi=x-ct,\ \eta=x+ct$ ; the congruence identity $\bar B^2-\bar A\bar C=J^2(B^2-AC)$ ; the Tricomi type-change $\Delta=-y$ with characteristics $x\mp\tfrac23(-y)^{3/2}=\text{const}$ ; the Hadamard solution $u=n^{-2}\sinh(ny)\sin(nx)$ and its blow-up at fixed $y>0$ ; and the worked-example roots $m=3,1$ with offset $5.33$ at $y=-4$ .

Prerequisites. Quadratic Forms and Eigenvalues · Partial Derivatives and the Chain Rule · Conic Sections. Leads to. The Wave Equation · The Heat (Diffusion) Equation · Laplace's and Poisson's Equations · Boundary Conditions and Well-Posedness · Method of Characteristics · d'Alembert's Solution and Characteristics of the Wave Equation.

Ask the tutor: Hand me any second-order PDE — even one with position-dependent coefficients — and I will classify it, find its characteristics, reduce it to canonical form, and tell you which boundary or initial data make it well-posed.

Why does only the principal symbol decide the type?

Because the type is a statement about the highest derivatives, which dominate the local behaviour and the characteristics. Substitute a rapidly oscillating trial solution $e^{i(\xi x+\eta y)/\epsilon}$ and let $\epsilon\to 0$ : second derivatives scale as $\epsilon^{-2}$ , first as $\epsilon^{-1}$ , the undifferentiated term as $\epsilon^0$ , so only $A\xi^2+2B\xi\eta+C\eta^2$ survives. Lower-order terms are invisible to that leading form — which is exactly why adding $u_x$ cannot turn a parabola into a hyperbola.

What actually is a characteristic, physically?

A characteristic is a curve along which the equation fails to determine the second derivatives from Cauchy data — the value and normal derivative of $u$ leave the second normal derivative undetermined. Physically these are the tracks along which information and discontinuities travel. In the wave equation they are the light-cone lines $x\pm ct=\text{const}$ , so a kink in the initial data propagates along them at exactly $c$ , and a point feels only the data in the interval its two characteristics cut out — its domain of dependence. Elliptic equations have no real characteristics, which is exactly why they permit no propagation and instead tie every interior point to the whole boundary at once.

How is this the same algebra as conic sections?

Set the principal symbol $A\xi^2+2B\xi\eta+C\eta^2$ equal to a constant: that is a conic in the $(\xi,\eta)$ plane, and $B^2-AC$ decides whether it is an ellipse (negative), a parabola (zero), or a hyperbola (positive) — the identical test for $Ax^2+2Bxy+Cy^2=1$ . Replacing $x,y$ by $\xi,\eta$ turns the conic into the principal symbol, so the geometric names transfer verbatim. A conic's shape and a field equation's physics are the same invariant seen twice.

Does the type tell me what boundary or initial data to use?

Yes — that is its main practical payoff. Elliptic equations are boundary-value problems: specify $u$ (Dirichlet) or its normal derivative (Neumann) on a closed surface, as for electrostatics. Parabolic equations need an initial profile $u(x,0)$ plus boundary conditions for all later time, as for a cooling rod. Hyperbolic equations are Cauchy problems: give $u$ and $u_t$ on an initial non-characteristic surface plus boundary conditions, as for a plucked string. Mismatching them is Hadamard's disaster — Cauchy data on an elliptic equation gives solutions that exist for hardly any data and depend discontinuously on it.

Can one equation be of more than one type?

Yes, whenever the leading coefficients depend on position. The discriminant $\Delta=B^2-AC$ is then a function whose sign can change across the domain, so the equation is genuinely of different types in different regions. The Tricomi equation $y\,u_{xx}+u_{yy}=0$ has $\Delta=-y$ : elliptic where $y>0$ , hyperbolic where $y<0$ , parabolic on the line between. This is the mathematics of transonic flow, where a subsonic (elliptic) region and a supersonic (hyperbolic) region coexist around a wing, split by the sonic line. Classifying such equations pointwise, and the analysis near the type-change line, remains a hard active subject.

What are parabolic and elliptic canonical forms, and why is only one derived here?

By choosing coordinates adapted to its characteristics, every second-order equation transforms into one of three normal forms: hyperbolic $u_{\xi\eta}=\text{lower order}$ , elliptic $u_{\xi\xi}+u_{\eta\eta}=\text{lower order}$ , parabolic $u_{\eta\eta}=\text{lower order}$ . The hyperbolic reduction is worked above because its two distinct real characteristics make the change transparent. The elliptic case uses complex-conjugate characteristics — take their real and imaginary parts as the new coordinates, turning the mixed form into the Laplacian. The parabolic case has one characteristic family, so one coordinate is a characteristic and the other any transverse coordinate, leaving a single second derivative — the heat-equation shape.

Why does the wave equation have a finite propagation speed but the heat equation an infinite one?

It comes straight from the characteristics. The hyperbolic wave equation has two real characteristics of finite slope $\pm 1/c$ , so a signal is confined to the light cone and a point at time $t$ feels only the initial data within a distance $ct$ — a domain of dependence of finite width $2ct$ . The parabolic heat equation has a single degenerate characteristic family (the lines $t=\text{const}$ ), so there is no finite-speed barrier: its Gaussian heat kernel is strictly positive everywhere for any $t>0$ , so heating one point instantly warms — by an exponentially tiny amount — everywhere. The infinite speed is an idealisation of diffusion, and a direct fingerprint of the vanishing discriminant.