Well-posed problem

Remarks

The choice of the spaces $X$ and $Y$ and of their norms is part of the problem. A problem can be well posed for one choice and ill posed for another. Example: the Dirichlet problem for the Laplace equation on the closed unit disc $\overline{B}$, with $X = C(\partial B)$ and $Y = C(\overline{B})$ both with the supremum norm. The Poisson integral gives a solution for each boundary function $\varphi$. If $u$ and $v$ solve the problem with boundary values $\varphi$ and $\psi$, the maximum principle applied to $\pm(u - v)$ gives $\max_{\overline{B}} \abs{u - v} \le \max_{\partial B} \abs{\varphi - \psi}$. This bound gives both uniqueness and continuity, so the problem is well posed. Non-example: the Cauchy problem for the Laplace equation fails clause 3; see Hadamard's example. The backward heat equation also fails clause 3, and on bounded domains it fails clause 1 for most data.