satisfies equation (1.5) is called the strong solution. The generalized formulation has the following important properties:
1 The exact solution, denoted by , exists for all data that satisfy the conditions where α and β are real numbers, and f is such that satisfies the definitive properties of linear forms listed in Section A.1.2 for all . Note that κ, c and f can be discontinuous functions.
2 The exact solution is unique in the energy space, see Theorem 1.1.
3 If the data are sufficiently smooth for the strong solution to exist then the strong and weak solutions are the same.
4 This formulation makes it possible to find approximations to with arbitrary accuracy. This will be addressed in detail in subsequent sections.
Exercise 1.2 Assume that
Exercise 1.3 Consider the sequence of functions
illustrated in Fig. 1.2. Show that
This exercise illustrates that restriction imposed on
Exercise 1.4 Show that
Figure 1.2 Exercise 1.3: The function
Remark 1.1
(1.35)
1.2.2 The principle of minimum potential energy
Theorem 1.2 The function
(1.36)
on the space
Proof: For any
(1.37)
