Well-posed problem

The problem of determining a solution in a metric space (with distance ) from initial data in a metric space (with distance ), satisfying the following conditions: a) for any there exists a solution ; b) the solution is uniquely defined; c) the problem is stable with respect to the spaces : For any there exists a such that, for any , the inequality implies , where , .

Problems not satisfying one of these conditions for well-posedness are called ill-posed problems.

Comments

The term "well-posed" (also properly posed or correctly set) was coined by the French mathematician J. Hadamard at the beginning of the 19th century [a1]. In particular, he stressed the importance of continuous dependence of solutions on the data (i.e. property 3). Practical problems (e.g. in hydrodynamics, seismology) lead not seldom to formulations that are ill-posed (cf. Ill-posed problems).

References

[a1]	J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952) (Translated from French)
[a2]	P.R. Garabedian, "Partial differential equations" , Wiley (1964)