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Session 36. Topology in Functional Analysis |
Parameter-dependence of ODE's |
| Helge Glöckner, Universität Paderborn, Germany |
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Let \(G\) be a Lie group modelled on a locally convex space,
with identity element \(e\) and Lie algebra \({\mathfrak g}\).
We say that \(G\) is \(C^0\)-semiregular
if every continuous curve \(\gamma\colon [0,1]\to {\mathfrak g}\)
arises as the left logarithmic derivative of a
(necessarily unique) \(C^1\)-curve \(\eta=\eta_\gamma\colon [0,1]\to G\)
with \(\eta(0)=e\). If, moreover, the map
\[
\text{evol}\colon C([0,1],{\mathfrak g})\to G, \;\;\;
\gamma\mapsto \eta_\gamma(1)
\]
is smooth, then \(G\) is called \(C^0\)-regular.
Thus, we are interested in the existence of solutions
to certain initial value problems on a Lie group,
and their dependence on parameters.
I show that continuous dependence frequently
entails smooth dependence.
To this end, I first observe
that \(\text{evol}\) is continuous if
and only if it is continuous at \(0\).
The main result then reads:
Theorem: If \(G\) is \(C^0\)-semiregular, \(\mbox{evol}\) is continuous at \(0\) and the smooth homomorphisms from \(G\) to \(C^0\)-regular Lie groups separate points on \(G\), then \(G\) is \(C^0\)-regular.
As an application, consider a finite-dimensional Lie group \(H\)
with compact Lie algebra \({\mathfrak h}\).
As recently shown in a Master's thesis by Timm Pieper
(Paderborn), there is a
Lie group \(C^\infty_{\mathcal W}({\mathbb R},H)\)
of certain \(H\)-valued smooth maps on the line
which is modelled on the weighted function space
\(C^\infty_{\mathcal W}({\mathbb R},{\mathfrak h})\),
for the set \({\mathcal W}\) of all weight functions
\(f_a\colon {\mathbb R}\to $\,$ ]0,\infty[\), \(f_a(t):=e^{-a|t|}\)
with \(a>0\).
I'll explain how the above
theorem can be used to see that
the Lie group
\(C^\infty_{\mathcal W}({\mathbb R},H)\)
is \(C^0\)-regular.
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