ABSTRACT
We recall in our use of the Lumer-Phillips generation theorem (Theorem 3.4) with several examples a major task was verifying the range statement which is equivalent to solving equations of the form (λ0 − A)φ = ψ for any given ψ ∈ X. In this section we introduce and discuss fundamental mathematical concepts, sesquilinear forms, that will provide an important tool in treating succinctly such range statements. We then will discuss several forms of the celebrated LaxMilgram theorem that readily yield the desired range results needed in the Lumer-Phillips generation theorem for numerous examples. To motivate and illustrate our discussions, we introduce another example which is ubiquitous in structural applications and is also ideal for use in illustrating our weak formulations (as opposed to the semigroup formulations) of equations that follow in subsequent sections of this book.
We consider the beam equation given by
