8.4 Existence of expansions using Fuchsian methods, T3-case
In [54
], Kichenassamy and Rendall proved the existence of expansions in the real analytic setting. In
other words, they proved that, given real analytic
with
, there is a unique solution
to Equations (16) – (17) with expansions of the form of Equations (37) – (38). Note that the number of
functions that are freely specifiable in the expansions coincides with the number of functions that need to
be specified in order to obtain a unique solution to the initial value problem corresponding to
Equations (16) – (17). For reasons mentioned in Section 8.2, the expansions suffer from a potential
consistency problem in the case of
and
. However, in [54
] it was proven that if
is
constant, the condition on
can be relaxed to
. The regularity condition of [54
] was relaxed to
smoothness in [68
].