5.1 Weak Derivatives and Weak Solutions

Consider the Cauchy problem ( CW). It was noted in Section 3.1 that to have a solution u ? C ² (R R ⁺) of ( CW) we require ? ? C ² (R) and ? ? C ¹ (R). If the last assumptions are not satisfied then the solution given by D Alembert formula is not a classical solution. How to justify the meaning of a solution in this case? There exist two main approaches. One is to introduce the so called weak derivatives so that the wave equation is satisfied in a form of integral identity. The other is the sequential approach. Consider approximating problems with smooth data ( ? _k, ? _k) ? C ² ( R) C ¹ (R). It is possible to define a weak ( generalized) solution of the problem by passing to the limit in L ² spaces of corresponding solutions u _k. We prove that in some sense these two approaches are equivalent.

Let L ² ( ?) be the usual Lebesgue space of square integrable functions u : ? ? R, where ? is a measurable domain in R ⁿ. L ² ( ?) is a Banach space with a norm

Denote by ( ?) the space of test functions, i.e. all functions ? ( x)