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Hadamard defined a PDE problem as well-posed if it satisfies three conditions: existence (a solution exists for given data), uniqueness (the solution is the only one), and continuous dependence on data (small changes in the data produce small changes in the solution). A problem failing any condition is ill-posed. Well-posedness determines whether a mathematical model is physically meaningful and computationally tractable. The classification of PDEs is intimately linked to well-posedness: each PDE type has specific data requirements (boundary conditions, initial conditions) that make the problem well-posed.
Jacques Hadamard introduced the concept of well-posedness in 1902, identifying the three properties that a PDE problem must have to be physically and computationally meaningful. Existence guarantees that the mathematical model has a solution—that the equations are not internally contradictory. Uniqueness ensures that the model makes definite predictions—the physical situation described determines a single outcome. Continuous dependence on data means the model is robust—since physical measurements always contain errors, the predicted solution must not be arbitrarily sensitive to these errors.
The concept of well-posedness is deeply linked to the classification of PDEs. Each type of equation is well-posed with specific types of auxiliary conditions. Elliptic equations like Laplace's equation are well-posed as boundary value problems (specifying u on all of ∂Ω) but ill-posed as initial value problems. Hyperbolic equations like the wave equation are well-posed as initial value problems (specifying u and u_t at t = 0) but require exactly the right amount of boundary data—too much or too little leads to ill-posedness. Parabolic equations like the heat equation are well-posed forward in time but ill-posed backward.
Hadamard's counterexample for the Cauchy problem of Laplace's equation is a landmark in PDE theory. He showed that specifying u and ∂u/∂n on a portion of the boundary does not determine u continuously throughout the domain: oscillatory data with amplitude 1/n produces solutions that grow like en. This is not a technicality—it means that no amount of measurement precision on the boundary can reliably predict the solution in the interior from Cauchy data alone.
Despite Hadamard's original intent that ill-posed problems should be avoided, the 20th century revealed that many important practical problems are ill-posed: inverse problems (determining an internal structure from boundary measurements), data assimilation (combining models with noisy observations), and backward-in-time problems all fail continuous dependence. The theory of regularization, developed by Tikhonov and others, provides systematic methods for extracting useful approximate solutions from ill-posed problems by adding stabilizing constraints. Understanding well-posedness remains essential: it tells us which problems can be solved directly and which require the additional machinery of regularization theory.