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.