Well-posed problem

The problem of determining a solution $z=R(u)$ in a metric space $Z$ (with distance $\rho_Z({\cdot},{\cdot})$) from initial data $u$ in a metric space $U$ (with distance $\rho_U({\cdot},{\cdot})$), satisfying the following conditions: a) for any $u \in U$ there exists a solution $z \in Z$; b) the solution is uniquely defined; c) the problem is stable with respect to the spaces $(Z,U)$: For any $\epsilon>0$ there exists a $\delta(\epsilon)>0$ such that, for any $u_1,u_2 \in U$, the inequality $\rho_U(u_1,u_2) < \delta(\epsilon)$ implies $\rho_Z(z_1,z_2) < \epsilon$, where $z_1 = R(u_1)$, $z_2 = R(u_2)$.

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) Zbl 0049.34805
[a2]	P.R. Garabedian, "Partial differential equations" , Wiley (1964) Zbl 0124.30501