at first, I'm quite sure it's not necessary to pay too much attention to the way the Operator is defined, it's rather important which spaces to choose to obtain a self-adjoint operator
I've got a problem with underständing the definition of the spaces of domain and range. Maybe someone is familiar with problems like that or even know it. In addition I should note that the following is part of a book (Joel Smoller-Shock Waves and Reaction-Diffusion Equations 106/107)
Okay, here it begins:
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Given is the operator $Pu = Au + au= \sum_{i,j=1}^N (a_{ij}(x)u_{x_{i}})_{x_{j}}+a(x)u$ defined in a bounded domain $\Omega \subset \mathbb{R}^n$, $\partial\Omega$ smooth. We assume the coefficients of P are smooth and $a_{ij}=a_{ji}$ in $\Omega$ Furthermore we assume A is strong elliptic in \Omega,in the sense that there is an $\alpha >0$ such that $\sum_{i,j=1}^Na_{ij}(x)\xi_i \xi_j>=\alpha |\xi|^2 , x \in \Omega$
We consider the operator P together with homogeneous boundary conditions of the form $\alpha(x)\frac{du}{dn}+\beta(x)u=0$ on $\partial\Omega$ where $\alpha(x)\geq 0 , \beta(x)\geq 0 , \alpha(x)^2 + \beta(x)^2 \ne 0$ We regard $P$ as an Operator acting on functions in $W^2(\Omega)$ which satisfy the boundary conditions on $\partial\Omega$, into $L_2(\Omega)$
The statement now is that P is a self-adjoint operator!
But this is my problem, to check the self-adjointness. Formally I need a hilbert-space $H$ and an operator $L:D(L)\subset H \rightarrow H$ defined on a dense subset $D(L)\subset H$
But in this case, I've got two different hilbertspaces where D(P) is (probably) a dense subset of $W^2(\Omega)$ but dense in $L^2(\Omega)$.
I hope someone could help me with that.
Showing an operator such as $P$ is symmetric on its domain $\mathcal{D}(P)$ is the first place to start. All selfadjoint operators are symmetric, but not necessarily the other way around. So, first show that $$ (Pf,g)=(f,Pg),\;\;\; f,g \in \mathcal{D}(P). $$ The inner-product should be the $L^{2}(\Omega)$ inner-product. The domain $\mathcal{D}(P)$ consists of all $u\in W^{2}(\Omega)$ which satisfy the boundary condition $\alpha \frac{\partial u}{\partial n}+\beta u=0$. And it is useful to show $P$ is semibounded if it is; that is, is there a constant $m$ such that $$ (Pf,f) \ge m\|f\|^{2},\;\;\; f \in \mathcal{D}(P)? $$ It's not important to establish that $\mathcal{D}(P)$ is dense in $W^{2}(\Omega)$--in fact, that won't be true. But it is important to show that $\mathcal{D}(P)$ is dense in $L^{2}(\Omega)$. If $P$ is not densely-defined in $L^{2}(\Omega)$, then an adjoint $P^{\star}$ isn't well-defined, and selfadjoint has no meaning. The underlying Hilbert space $H$ is $L^{2}(\Omega)$. The fact that $W^{2}(\Omega)$ is a Hilbert space is not directly relevant; you're just trying to find a domain $\mathcal{D}(P)$ on which $P$ will be a densely-defined selfadjoint linear operator. So, the setting is $$ P : \mathcal{D}(P)\subset L^{2}(\Omega)\rightarrow L^{2}(\Omega), $$ and you want to choose that domain so that $P=P^{\star}$, where the adjoint is taken in $L^{2}(\Omega)$.
It may seem hard to prove that the domain is dense. However, if you can show that $P$ is symmetric and that $P-\lambda I$, $P-\overline{\lambda}I$ are surjective for some $\lambda \in \mathbb{C}\setminus\mathbb{R}$, then the domain must be dense and $P$ must be selfadjoint. If $P$ is bounded below by some real $m$ as above, one need only verify that $P+\lambda I$ is surjection for some real $\lambda > -m$. Either verification usually comes down to classical solvability for the PDE.