Commit c9c1c228 authored by Jannes Bantje's avatar Jannes Bantje

some work towards complete diff ops

parent 747567ee
Pipeline #8016 passed with stages
in 58 seconds
......@@ -149,7 +149,7 @@ There is a coordinate-free description as well:
\[
\benbrace*{\mathd,f} \omega = \mathd(f \omega) - f(\mathd \omega) = (\mathd f) \wedge \omega + f \mathd \omega - f \mathd \omega = (\mathd f) \wedge \omega
\]
Again via \cref{thm:diff_op_coordfree} we see that $\mathd$ is a differential operator of order one.
Again via \cref{thm:diff_op_coordfree} we see that $\mathd$ is a differential operator of order 1.
\item Let $E \to M$ be an arbitrary vector bundle with a \Index{connection} $\nabla \colon \Gamma(M,E) \to \Gamma(M, \Tang^*M \otimes E)$, i.e. $\nabla$ is linear and fulfils $\nabla(f s)= \mathd f \otimes s + f \nabla s$ for $f \in C^\infty(M)$ and $s \in \Gamma(M,E)$.
In particular
\[
......@@ -298,7 +298,7 @@ Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\
Note that the closure $\overline{D}$ is an extension of $D$ in general and that $D^*$ is an extension of $\overline{D}$ in the symmetric case.
As the following example shows, all three operators may be different:
\begin{example}
\begin{example}\label{ex:max_domain_neq_min_domain}
Consider the open $1$-manifold $M=(0,1)$ (with the usual Riemannian metric) and $D= i \diffd{}{x}$ (we need the factor $i$ here for $D$ to be symmetric, use integration by parts to see this).
The Hilbert space we get is $L^2[0,1]$.
......@@ -310,15 +310,17 @@ As the following example shows, all three operators may be different:
This example already suggests, that non-selfadjointness is in some way connected to \enquote{boundary conditions} on a non-compact manifold $M$.
The same phenomenon shows up in the following statement:
\begin{lemma}
\begin{lemma}\label{lem:symmetric_ess_sa}
Let $D$ be a symmetric differential operator and $u \in L^2(M,E)$ \emph{compactly supported}.
Then $u$ belongs to the minimal domain of $D$ if and only if it belongs to the maximal domain.
\end{lemma}
In particular we see, that every symmetric differential operator on a closed manifold is essentially selfadjoint.
More generally every compactly supported, symmetric differential operator on an open manifold is essentially selfadjoint.
\begin{corollary}
Every symmetric differential operator on a closed manifold is essentially selfadjoint.
More generally every compactly supported, symmetric differential operator on an open manifold is essentially selfadjoint.
\end{corollary}
\begin{proof}
\begin{proof}[Proof of \cref{lem:symmetric_ess_sa}]
Let $K$ be a compact subset of $M$.
We need \emph{Friedrichs' mollifiers} to proceed:
A local variant is discussed in \cref{prop:friedrichs}, which can be globalised via the usual arguments to operators $F_t \colon L^2(K,E) \to L^2(M,E)$ such that
......@@ -333,11 +335,20 @@ More generally every compactly supported, symmetric differential operator on an
\[
D F_t u = F_t D u + \benbrace*{D,F_t} u
\]
By \cref{rmk:max_domain_distribution} and (a) we know, that the first summand is uniformly bounded ; the second one is uniformly bounded by (d).
By \cref{rmk:max_domain_distribution} and (a) we know, that the first summand is uniformly bounded; the second one is uniformly bounded by (d).
Considering $t=\frac{1}{n}$ we get a sequence of elements $u_n$ in the domain of $D$ converging to $u$ such that $\norm*{D u_n}$ is uniformly bounded.
Hence \cref{lem:charac_minimal_domain} implies, that $u$ is in the minimal domain of $D$.
\end{proof}
As seen in \cref{ex:max_domain_neq_min_domain} the selfadjointness of differential operators on open manifolds is a more delicate matter.
We need to look at multiplication operators.
\begin{definition}
Let $D$ be a differential operator on $M$.
Then $M$ is called \Index{complete} for $D$ if there is a smooth proper function $g \colon M \to \mathbb{R}$ such that $\benbrace*{D,\rho(g)}$ is a bounded Hilbert space operator.
\end{definition}
\todo[inline]{include wave operator stuff as well?}
\begin{lemma}\label{lem:adjoint_elliptic}
If $D$ is a symmetric elliptic operator, then its adjoint is elliptic as well.
......@@ -345,8 +356,6 @@ More generally every compactly supported, symmetric differential operator on an
\begin{proof}
\todo{add proof}
\end{proof}
\todo[inline]{include wave operator stuff as well?}
% section hilbert_space_techniques_and_the_formal_adjoint (end)
\section{Analysis of Elliptic Differential Operators} % (fold)
......
Markdown is supported
0%
or
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment