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-
posedness.

Wed Feb 4 14:46:04 EST 1998