Orateur
Elmar Schrohe
Description
We model a conic manifold by a manifold $\mathbb B$ with boundary
$\partial \mathbb B=:X$.
In a collar neighborhood we introduce coordinates $(t,x)$,
where $t$ is the distance to $\partial \mathbb B$ and $x$ the variable in
$X$.
A cone differential operator of order $\mu$ is an operator
$A: C^\infty_c(\mathbb B^\circ)\to C^\infty_c(\mathbb B$ that can be written
near $\partial \mathbb B$ in the form
$$A=t^{-\mu}\sum_{k=0}^\mu a_k(t)(-t\partial_t)^k\quad \text{with }a_k\in C^\infty([0,1)\mathrm{Diff}^{\mu-k}(X)).
$$
We consider an extension of $A$ in a weighted cone Sobolev space $\mathcal H^{s,\gamma}_p(\mathbb B)$ with domain $\mathcal D(A) = \mathcal H^{s+\mu,\gamma+\mu}_p(\mathbb B)\oplus \mathcal E$, where $\mathcal E$ is a space of asymptotics functions.
Given a sector
$$\Lambda_\theta= \{re^{i\phi}: r\ge0, \theta\le\phi\le2\pi-\theta\}, \quad 0<\theta<\pi,$$
we show that any extension of $A$ which is parameter-elliptic with respect to
$\Lambda_\theta$ has a bounded $H^\infty$-calculus on $\mathbb C\setminus \Lambda_\theta.$
Parameter-ellipticity with respect to $\Lambda_\theta$ here requires the following
\begin{enumerate}
\item Denote by $\sigma_\psi^\mu(A)$ the principal symbol of $A$. Then $\sigma_\psi^\mu(A)-\lambda$ is invertible for $\lambda \in \Lambda_\theta$, even up to the boundary, if one takes into account the degeneracy.
\item The principal conormal symbol $\sigma_M^\mu(A)(z)$ is invertible for all $z\in \mathbb C$ with ${\rm Re}\, z= \frac{n+1}2-\gamma -\mu $ or ${\rm Re}\, z= \frac{n+1}2-\gamma$.
\item $\Lambda_\theta$ is a sector of minimal growth for the model cone operator
$$\widehat A= t^{-\mu}\sum_{k=0}^\mu a_k(0)(-t\partial_t)^k$$
acting on cone Sobolev spaces over he infinite cone.
\end{enumerate}
Applications concern the Laplacian and the porous medium equation on manifolds with warped conical singularities.
(Joint work with J\"org Seiler (Torino))
