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Second-order linear PDEs in two variables are classified as elliptic, parabolic, or hyperbolic based on the discriminant of their principal part, analogous to how the discriminant classifies conic sections. The classification determines the qualitative behavior of solutions: elliptic equations (like Laplace's equation) describe steady-state phenomena, parabolic equations (like the heat equation) describe diffusion processes, and hyperbolic equations (like the wave equation) describe propagation phenomena. This trichotomy governs which boundary and initial conditions produce well-posed problems.
The classification of second-order linear PDEs is one of the most important organizational principles in the theory. Given a general second-order PDE Au_xx + 2Bu_xy + Cu_yy + (lower order terms) = 0, the quantity D = B² - AC determines its type: elliptic if D < 0, parabolic if D = 0, and hyperbolic if D > 0. This is not merely a formal exercise—it reflects deep differences in the physical phenomena described and the mathematical behavior of solutions.
Elliptic PDEs, typified by Laplace's equation Δu = 0, describe steady-state or equilibrium phenomena. Their solutions are infinitely smooth in the interior of the domain (a regularity property), satisfy maximum principles (the solution achieves its extremes on the boundary), and are completely determined by boundary data alone. There are no characteristic curves, and information propagates in all directions simultaneously.
Parabolic PDEs, typified by the heat equation u_t = kΔu, describe diffusive processes that evolve in time. They have a single family of characteristic surfaces (the level sets of t), and information propagates with infinite speed—a disturbance at one point is instantly felt everywhere, though its effect decays rapidly with distance. Solutions smooth out immediately: even rough initial data produces smooth solutions for any t > 0.
Hyperbolic PDEs, typified by the wave equation u_tt = c²Δu, describe propagation phenomena. They have two families of real characteristic curves along which information travels at finite speed, creating a domain of dependence and a domain of influence for each point. Unlike elliptic and parabolic equations, hyperbolic equations preserve singularities in the initial data—waves maintain their shape as they propagate. This finite propagation speed is the defining physical feature of hyperbolic problems.
The classification extends to higher dimensions, though the analysis becomes more nuanced. For a PDE with n spatial variables, the principal symbol is an n×n matrix, and the equation is elliptic if all eigenvalues have the same sign, hyperbolic if exactly one differs, and parabolic if at least one eigenvalue is zero. Systems of PDEs and variable-coefficient equations can change type across the domain, leading to rich and challenging mathematical problems at the boundaries between regions of different type.