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6 Well Posed PDE Problems     continued...

To illustrate that boundary value problems, not initial value problems, are the appropriate setting for elliptic PDE problems, we present the following example due to Hadamard. To view this problem as an initial value problem, one should think of y as a time variable. Consider the initial value problem

For and , it is clear that the corresponding solution to the above initial value problem is . For the case and , it is easy to verify that the corresponding solution is

Observe that the functions and are identical and that

uniformly in x. Thus, we see that the data of the two problems, , and , , can be made arbitrarily close. But, if we compare the two solutions at , then we obtain

For y positive, approaches infinity faster than , as n goes to infinity. Therefore, we conclude that

illustrating that as the data for the two problems becomes more alike, the solutions become increasingly different. This is what is meant by failure of the solution to depend continuously on the problem data.