We now examine Hadamard's example of an ill-posed problem. Consider
Then is the unique solution. (Easy).
But now consider large n. We have for any Sobolev
norm because the beats all polynomials in n (from
any finite order of
derivatives). Hence we can choose n so large that the boundary
condition is as close to 0
as we please. But the solution grows large at any fixed
line y = constant as ; hence .
( solves , uniquely).
So this problem does not depend continuously on the data, and
hence is not well-posed.
Therefore it is not something we can safely ignore, well-