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Hadamard's example

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.

Robert Corless
Wed Feb 4 14:46:04 EST 1998