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As is well known, a problem is said to be well-posed in the sense of
Hadamard when a unique solution exists and depends continuously upon the
data. The definition is made precise by stipulating not only the function
spaces in which the solution and data are to lie but also the measures and
notion of continuity. A problem that is not well-posed is said to be ill-posed.
Although nineteenth-century mathematicians contributed to the early
study of ill-posed problems, it is generally agreed that the subject came to
prominence only after Hadamard had formulated his well-known definition.
Unfortunately, he developed an adverse view of the subject which, on becoming widely accepted, had the effect of inhibiting further study. His objections
were grounded in his celebrated counterexample of the Cauchy problem for
Laplace's equation. In order for there to be global existence of the solution
Hadamard demonstrated that the Cauchy data must satisfy a certain compatibility relation but even in the unlikely event of the relation being satisfied he
further showed that the solution in general does not depend continuously on
the data. Such behaviour convinced Hadamard that ill-posed problems lacked
physical relevance and hence should be ignored. This became the prevailing
attitude, and consequently, in partial differential equations at least, activity
became confined to the standard initial boundary value problems. It was only
the growing insistence for a precise theoretical understanding from the applied
sciences, principally geophysics and computing, that rekindled mathematical
interest.
As is well known, a problem is said to be well-posed in the sense of
Hadamard when a unique solution exists and depends continuously upon the
data. The definition is made precise by stipulating not only the function
spaces in which the solution and data are to lie but also the measures and
notion of continuity. A problem that is not well-posed is said to be ill-posed.
Although nineteenth-century mathematicians contributed to the early
study of ill-posed problems, it is generally agreed that the subject came to
prominence only after Hadamard had formulated his well-known definition.
Unfortunately, he developed an adverse view of the subject which, on becoming widely accepted, had the effect of inhibiting further study. His objections
were grounded in his celebrated counterexample of the Cauchy problem for
Laplace's equation. In order for there to be global existence of the solution
Hadamard demonstrated that the Cauchy data must satisfy a certain compatibility relation but even in the unlikely event of the relation being satisfied he
further showed that the solution in general does not depend continuously on
the data. Such behaviour convinced Hadamard that ill-posed problems lacked
physical relevance and hence should be ignored. This became the prevailing
attitude, and consequently, in partial differential equations at least, activity
became confined to the standard initial boundary value problems. It was only
the growing insistence for a precise theoretical understanding from the applied
sciences, principally geophysics and computing, that rekindled mathematical
interest.
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