Wave equations with Dirichlet boundary condition

Question

In his book chap 4, Fattorini wrote :

we consider the equation $$u''(t)=A(\beta)u(t)$$ where $$Au= \sum_{i,j=1}^m \frac{\partial}{\partial x_i}\left(a_{i,j}(x) \frac{\partial u}{\partial x_j}\right)+\sum_{i=1}^m b_i(x) \frac{\partial u}{\partial x_i}+c(x)u$$ $x=(x_1,x_2,\cdots,x_m)$ and $a_{ij}(x),b_j(x),c_j(x)$ are defined in a domain $\Omega \subset \mathbb{R}^m$. $A(\beta)$ denotes the restriction of $A$ obtained by means of a boundary condition $\beta$ at the boundary $\partial\Omega$ of the form $$u(x)=0 \quad (x \in \partial \Omega).$$ I didn't understand the part that $A(\beta)$ is the restriction of $A$... what is $A(\beta)$ explicitly in term of $A$.

Answer 1

By definition, $Au$ is a differential operator. It is defined by the expression determining $(Au)(x)$ in terms of coefficient functions and derivatives of $u$ at $x$. Here, it does not make sense to talk about boundary conditions.

One can $A$ also interpret as a mapping between function spaces. Then the boundary conditions would be encoded in the domain of $A$. In this way, $A$ may change if different boundary conditions are applied. That is, the domain of definition of $A$ changes.

Answer 2

The restriction (a.k.a realization) part means you just define (or restrict) the domain of the differential operator on a subspace. Of course, when you change the domain you change the operator. For instance, in the case of homogeneous Dirichlet condition $\beta u=\gamma u=0$. Therefore, $D(A(\beta))=H^1_0(\Omega)$ and $A(\beta)u=Au$. Similarly, for Neumann condition, you have $\beta u= \partial_n u=0$, $D(A)=\{u\in H^1(\Omega): \partial_n u=0\}$ and $A(\beta)u=Au$.