2
1. DIFFERENTIA L EQUATION S AN D THEI R GEOMETR Y
The Cauch y proble m fo r (1.1) admit s a simpl e geometri c formulation , whic h w e
now recall , [Ar2] , [G] . Th e Cauch y dat a ar e give n b y a C°° ma p 0 : U C M p _ 1 —»
R 2 p + 1 , o f ran k p - 1, suc h tha t
(1.8) 0(U) = iV
p
_x C M
2 p
,
where 7V
p
_i i s a (p l)-dimensiona l submanifol d embedde d i n M2
P
, an d
(1.9) 6*UJ = 0.
Given Cauch y dat a {11,9) a s above , th e Cauch y proble m i s t o find a C°° ma p
a : U x (a , 6) C M p - M 2 p + 1, o r ran k p , suc h tha t
(1.10) a(U x (a , b)) C M
2 p
, cr* ^ = 0 ,
and,
(1.11) a{s,t
o
) = 0(s),
for som e t o G (a, 6) an d fo r al l s G U.
The solutio n o f th e abov e Cauch y proble m i s o f cours e classical , an d base d o n
the constructio n o f a vecto r field Xp, calle d th e Cauchy characteristic vector field
of (1.1), whic h i s tangen t t o M ^ , an d suc h tha t th e ma p
(1.12) (7(3 , t) = ex p \
e(s)
((t - t
0
)XF),
solves the Cauch y problem . Th e vecto r field X
F
i s defined i n the followin g way , [Ar2] .
In th e tangen t spac e T
x
M2pi on e consider s th e (2p— l)-dimensiona l subspac e V2p_ i
defined b y
(1.13) V2
P
-i : ^
x
n T
x
M
2 p
.
By takin g th e skew-orthogona l complemen t o f V ^ - i wit h respec t t o Q, a t eac h poin t
x G M2
P
, on e obtain s a lin e field o n M.2
V
. Thi s lin e field i s precisel y spanne d b y
the vecto r field Xp, whos e expressio n i s give n b y
^—f I 9ui dx l % dui du dx i % du dui J '
We sa y tha t th e Cauch y dat a -/V
p
_i : = 0(U) i s non-characteristic i f th e transversal -
ity conditio n give n b y
(1.15) T
x
^
p
_
1xF
n { X | } = 0 ,
in T
X
M2P i s satisfie d a t ever y poin t x G Np-i. W e hav e
T H E O R E M 1.1If . the Cauchy data for (1.1) is non-characteristic, then the
Cauchy problem admits a unique local C°° solution given by
(1.16) a(s,t)=exp\
e{s)
{{t-t0)XF).
The solutio n o f the Cauch y proble m thu s reduce s t o finding C°° function s whic h
are constan t alon g th e th e integra l curve s o f Xp, als o calle d characteristi c strips .
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