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# Continuous Dependence on Data for Solutions of Partial Differential Equations With a Prescribed Bound

## Abstract

Problems in partial differential equations usually require that a solution *u* be determined from certain data *f*. Ordinarily the data consist of the values of *u* and of a certain number of derivatives of u on a manifold *Φ*. For the classical problems which are *well posed* in the sense of Hadamard the solution u exists and is determined uniquely for all *f* of some class *C* _{ s }; moreover *u* depends continuously on *f* if suitable norms are used. Usually no solution *u* exists in a problem which is not well posed, even for *f* in C_{∞}. Moreover, even for those *f* for which *u* exists we have no continuous dependence of *u* on *f*. This is best illustrated by the classical example of Hadamard [1] of the Cauchy problem for the Laplace equation *u* _{ xx } *+u=*0, *u=*0, *u* _{ y } *=f*(*x*), for which no solution exists unless *f* is analytic. If *f*(*x*) = *n* ^{-2} cos *nx* we do have a solution *u* = *n* ^{-3} (cos *nx*)(sinh *ny*). However though *f* and its first and second derivatives tend to zero for *n* →∞ the corresponding solutions *u* tend to ∞ for *x* = 0, *y* ≠ 0.

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