Journal of Inequalities in Pure and
Applied Mathematics
NECESSARY AND SUFFICIENT CONDITION FOR EXISTENCE
AND UNIQUENESS OF THE SOLUTION OF CAUCHY PROBLEM
FOR HOLOMORPHIC FUCHSIAN OPERATORS
volume 2, issue 2, article 24,
2001.
MEKKI TERBECHE
Florida Institute of Technology,
Department of Mathematical Sciences,
Melbourne, FL 32901, USA
EMail: terbeche@hotmail.com
Received —-;
accepted 5 March, 2001.
Communicated by: R.P. Agarwal
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
022-01
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Abstract
In this paper a Cauchy problem for holomorphic differential operators of Fuchsian type is investigated. Using Ovcyannikov techniques and the method of
majorants, a necessary and sufficient condition for existence and uniqueness
of the solution of the problem under consideration is shown.
2000 Mathematics Subject Classification: 35A10, 58A99.
Key words: Banach algebra, Cauchy problem, Fuchsian characteristic polynomial,
Fuchsian differential operator, Fuchsian principal weight, holomorphic
differentiable manifold, holomorphic hypersurface, Fuchsian principal
weight, method of majorants, method of successive approximations,
principal symbol, and reduced Fuchsian weight.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Notations and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Majorants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Statement of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5
Formal Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6
Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
References
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
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1.
Introduction
We introduce the method of majorants [2], [5], and [8], which plays an important role for the Cauchy problem in proving the existence of a solution. This
method has been applied by many mathematicians, in particular [1], [3], and [4]
to study Cauchy problems related to differential operators that are a “natural”
generalization of ordinary differential operators of Fuchsian type, and to generalize the Goursat problem [8]. We also give a refinement of the method of
successive approximations as in the Ovcyannikov Theorem given in [7]. Combining these two methods, we shall prove the theorem [6].
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
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2.
Notations and Definitions
Let us denote
x = (x0 , x1 , . . . , xn ) ≡ (x0 , x0 ) ∈ R × Rn , where x0 = (x1 , . . . , xn ) ∈ Rn ,
ξ = (ξ0 , ξ1 , . . . , ξn ) ≡ (ξ0 , ξ 0 ) ∈ R × Rn , where ξ 0 = (ξ1 , . . . , ξn ) ∈ Rn ,
α = (α0 , α1 , . . . , αn ) ≡ (α0 , α0 ) ∈ N × Nn , where α0 = (α1 , . . . , αn ) ∈ Nn .
We use Schwartz’s notations
0
xα = xα0 0 xα1 1 · · · xαnn ≡ xα0 0 (x0 )α , |x|α = |x0 |α0 |x1 |α1 · · · |xn |αn
α! = α0 !α1 ! · · · αn !, |α| = α0 + α1 + · · · + αn ,
β ≤ α means βj ≤ αj for all j = 0, 1, . . . , n,
∂ |α|
∂
α
D =
≡ D0α0 D1α1 · · · Dnαn , where Dj =
, 0 ≤ j ≤ n.
α0 α1
α
n
∂x0 ∂x1 ...∂xn
∂xj
For k ∈ N, 0 ≤ k ≤ m,
max[0, α0 + 1 − (m − k)] ≡ [α0 + 1 − (m − k)]+ ,
m!
m
, Cq (j) = j(j − 1)...(j − q + 1),
=
k
(m − k)!k!
by convention C0 (j) = 1, and the gradient of ϕ with respect to x will be denoted
by
∂ϕ(x)
∂ϕ(x)
gradϕ(x) =
,...,
.
∂x0
∂xn
P
We denote a linear differential operator of order m, P (x; D) by |α|≤m aα (x)Dα .
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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Definition 2.1. Let E be an n + 1 dimensional holomorphic differentiable manifold. Let h be a holomorphic differentiable operator over E of order m0 in a,
and of order ≤ m0 near a. Let S be a holomorphic hypersurface of E containing a, let m be an integer ≥ m0 , and let ϕ be a local equation of S in some
neighborhood of a, that is, there exists an open neighborhood Ω of a such that:
∀x ∈ Ω, gradϕ(x) 6= 0, x ∈ Ω ∩ S ⇐⇒ ϕ(x) = 0.
If σ ∈ Z and Y is a holomorphic function on Ω, for x ∈ Ω\S, we denote by
hσm (Y )(x) = ϕσ−m (x)h(Y ϕm )(x)
σ
and by Hm
(x, ξ) the principal symbol of this differential operator.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
(i)
τh,S (a) = inf σ ∈ Z : ∀Y holomorphic function in a neighborhood
σ+1
Ω of a, ∀x ∈ Ω ∩ S, lim hm (Y )(x) = 0
x→b,x∈S
/
denotes the Fuchsian weight of h in a with respect to S.
(ii)
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∗
τh,S (a) = inf σ ∈ Z :
−m0
σ+1
lim ϕ
(x)H (x; gradϕ(x)) = 0
x→b,x∈S
/
is the Fuchsian principal weight of h in a with respect to S.
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(iii)
τ̃h,S (a) = inf σ ∈ Z : ∀Ω, ∀Y, ∀b ∈ Ω ∩ S,
lim
x→b,x∈S
/
[hσ+1
m (Y
)(x) − Y
(x)hσ+1
m (1)(x)]
=0
denotes the reduced Fuchsian weight of h in a with respect to S.
A differential operator h is said to be a Fuchsian operator of weight τ in a
with respect to S if the following assertions are valid:
∗
(H-0) τh,S
is finite and constant and equal τ near a ∈ S,
(H-1) τh,S (a) = τ ,
(H-2) τ̃h,S (a) ≤ τ − 1.
Mekki Terbeche
A Fuchsian characteristic polynomial is defined to be a polynomial C in λ of
holomorphic coefficients in y ∈ S by
C(λ, y) =
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
lim ϕτh,S (a)−λ (x)h(ϕλ )(x).
x→y,x∈S
/
Set
C1 (λ, y) = C(λ + τh,S (a), y), ∀λ ∈ C, ∀y ∈ S.
Remark 2.1. If we choose a local card for which ϕ(x) = x0 and a = (0, . . . , 0),
we get Baouendi-Goulaouic’s definitions [1].
Remark 2.2. The number τh,S (a) is independent on m which is greater or equal
to m0 .
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3.
Majorants
The majorants play an important role in the Cauchy method to prove the existence of the solution, where the problem consists of finding a majorant function
which converges.
Let α be a multi-index of Nn+1 and E be a C-Banach algebra, we define a
formal series in x by
X
xα
u(x) =
uα ,
α!
n+1
α∈N
where uα ∈ E.
We denote by E[[x]] the set of the formal series in x with coefficients in E.
Definition 3.1. Let u(x), v(x) ∈ E[[x]], and λ ∈ C. We define the following
operations in E[[x]] by
P
α
(a) u(x) + v(x) = u(x) = α∈Nn+1 (uα + vα ) xα! ,
P
α
(b) λu(x) = α∈Nn+1 (λuα ) xα! ,
P
P
α
(c) u(x)v(x) = α∈Nn+1 0≤β≤α αβ uβ vα−β xα! .
Definition 3.2. Let
u(x) =
X
uα
α∈Nn+1
xα
∈ E[[x]],
α!
and
U (x) =
X
α∈Nn+1
Uα
xα
∈ R[[x]]
α!
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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be two formal series. We say that U majorizes u, written U (x) u(x), provided Uα ≥ kuα k for all multi-indices α.
Definition 3.3. Let
u(x) =
X
uα
α∈Nn+1
xα
∈ E[[x]].
α!
We define the integro-differentiation of u(x) by
X
xα
Dµ u(x) =
uα+µ ,
α!
α≥[−µ]+
for µ ∈ Zn+1 , and [−µ]+ ≡ ([−µ0 ]+ , [−µ1 ]+ , . . . , [−µn ]+ ).
(a) Let A be a finite subset of Zn+1 , P (x; D) is said to be a formal integrodifferential operator over E[[x]] if for u(x) ∈ E[[x]],
X
P (x; D) =
aµ (x)Dµ u(x),
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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µ∈A
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where aµ (x) ∈ E[[x]].
(b) Let
P(x; D) =
X
Aµ (x)Dµ
and
X
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µ∈Zn+1
P (x; D) =
JJ
J
aµ (x)Dµ
µ∈Zn+1
be formal integro-differential operators over R[[x]] and E[[x]] respectively.
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We say P(x; D) majorizes P (x; D), written P(x; D) P (x; D), provided
Aµ (x) aµ (x) for all multi-indices µ ∈ Zn+1 .
Definition 3.4. Consider a family {uj }j∈J , uj ∈ E[[x]]. The family {uj }j∈J is
said to be summable if for any α ∈ Nn+1 , Jα = {j ∈ J : ujα 6= 0} is finite.
Theorem 3.1. Let v ∈ E[[y]], (y = (y1 , ..., ym )), V ∈ R[[y]] such that V (y) v(y). Let uj (x) ∈ E[[x]] for j = 1, . . . , m, uj0 = 0, and U j (x) ∈ R[[x]] for
j = 1, . . . , m, U0j = 0 such that U (x) u(x) for all j = 1, . . . , m. Then
V U 1 (x), . . . , U m (x) v u1 (x), . . . , um (x) .
Proof. See [7].
Definition 3.5. If u(x) ∈ E[[x]], we denote the domain of convergence of u by
)
(
α
X
|x|
d(u) = x : x ∈ Cn+1 , u(x) =
kuα k
<∞ .
α!
α≥0
Theorem 3.2. (majorants):
If U (x) u(x), then
d(U ) ⊂ d(u).
The above theorem is practical because, if the majorant series U (x) converges for |x| < r then u(x) converges for |x| < r. Let us construct a majorant
series through an example.
Let r = (r0 , r1 , ..., rn ) and let u(x) be a bounded holomorphic function on
the polydisc
Pr = {x : x ∈ Cn+1 , | xj |< rj , for all j = 0, 1, . . . , n}.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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Let M = sup ku(x)k, then it follows from Cauchy integral formula that kuα k <
x∈Pr
M
α!.
rα
If we let Uα =
M
α!,
rα
then U (x) majorizes u(x).
Theorem 3.3. Let ai , 0 ≤ i ≤ m,be holomorphic functions near the origin in
Cn that satisfy the following condition
m
X
Ci (j)ai (0) 6= 0, ∀j ∈ N, (am = 1),
i=0
then there exists a holomorphic function A near the origin in Cn such that
1
A(x0 )
P
, ∀j ∈ N
m
0
i=0 Ci (j)ai (x )
i=0 Ci (j)
Pm
Proof. If we write
Pm
Pm
Ci (j)
i=0
i=0 Ci (j)
Pm
P
=
·
m
0
i=0 Ci (j)ai (x )
i=0 Ci (j)ai (0) 1 −
1
Pm
Ci (j)(ai (0)−ai (x
i=0
Pm
i=0 Ci (j)ai (0)
Cm (j)
j→∞ Cm (j)am (0)
1
=
am (0)
= 1.
lim
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then we have
Pm
Ci (j)
lim Pm i=0
=
j→∞
i=0 Ci (j)ai (0)
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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Therefore, there exists a constant C ≥ 1 such that
Pm
C
(j)
i
i=0
Pm
≤ C, ∀j ∈ N.
i=0 Ci (j)ai (0)
Theorem 3.5.
Let B(x0 ) be a common majorant to ai (0) − ai (x0 ) for all i = 0, 1, . . . , m with
B(0) = 0, that is
Pm
Pm
Ci (j)(ai (0) − ai (x0 ))
Ci (j) 0 i=0P
i=0
B(x ) Pm
m
Ci (j)ai (0)
Ci (j)ai (0) i=0
i=0
CB(x0 ).
It follows from Theorem 3.1 that
1
1−
Pm
Choosing then A(x0 ) =
Ci (j)(ai (0)−ai (x0 ))
i=0
Pm
i=0 Ci (j)ai (0)
1
,
1−CB(x0 )
1
.
1 − CB(x0 )
the desired conclusion easily yields.
Corollary 3.4. Under the conditions of Theorem 3.3, there exist two positive
real numbers M > 0 and r > 0 such that
1
1
M
·
, ∀j ∈ N,
C1 (j, x0 )
(j + 1)m 1 − rt(x0 )
P
where t(x0 ) = m
i=0 xi .
Proof. The proof is similar to the proof of the Theorem 3.3, it suffices to observe
that
Pm
Ci (j)
lim Pm i=0
= 1,
j→∞
i=0 Ci (j)ai (0)
then apply the following theorem.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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If A(x0 ) is holomorphic near the origin in Cn , there exist M > 0 and R > 0
such that:
M
A(x0 ) .
R − t(x0 )
P
for all x0 ∈ {x0 : ni=0 | xi |< R}.
Proof. See [8].
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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4.
Statement of the Main Result
Theorem 4.1 (Main Theorem). Let h be a Fuchsian holomorphic differential
operator of weight τh,S (a) in a with respect to a holomorphic hypersurface S
passing through a, of a holomorphic differential manifold E of dimension n+1,
and ϕ a local equation of S in some neighborhood of a. Then the following
assertions are equivalent:
i) for all λ ≥ τh,S (a), C(λ, x0 ) 6= 0
ii) for all holomorphic functions f and v in a neighborhood of a, there exists
a unique holomorphic function u solving the Cauchy problem
(4.1)
h(u) = f
u − v = O(ϕτh,S (a) ).
If we choose a local card such that ϕ(x) = x0 and a = (0, ..., 0), then we
obtain the Baouendi-Goulaouic’s Theorem [1], if k = 0, we obtain CauchyKovalevskaya Theorem, and if k = 1, we obtain Hasegawa’s Theorem [3].
The following theorem gives a relationship between a Fuchsian operator of
arbitrary weight and a Fuchsian operator of weight zero.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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Theorem 4.2. Let h be a Fuchsian holomorphic differential operator of weight
τh,S (a) in a with respect to a holomorphic hypersurface S passing through a,
of a holomorphic differentiable manifold E of dimension n + 1, and ϕ a local
equation of S in some neighborhood of a. If we define the operator h1 by
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Y → h1 (Y ) = h(Y ϕτh,S (a) ),
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then h1 is a Fuchsian holomorphic differential operator of weight zero in a
relative to a holomorphic hypersurface S.
If C (respectively C1 ) denotes the Fuchsian polynomial characteristic of h
(respectively h1 ), then
C1 (λ, y) = C (λ + τh,S (a), y) , ∀λ ∈ C, ∀y ∈ S.
Proof.
1. We look for the Fuchsian weight of h1 in a with respect to S.
Let m ≥ m0 , where m0 is the order of h, then
ϕσ+1−m (x)h1 (Y ϕm )(x) = ϕσ+1−m (x)h(Y ϕm+τh,S (a) )(x)
= ϕ(σ+τh,S (a)+1)−(m+τh,S (a)) (x)h(Y ϕm+τh,S (a) )(x),
consequently τh,S (a) = 0.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
2. We look for the principal Fuchsian weight of h1 in a relative to S.
In the local card ϕ(x) = x0 and a = (0, . . . , 0), it is the exponent of x0 in
the coefficient of D0m of h1 . In this local card
0 −k
P1 (u) = P (xm
u)
0
k m m0 −k
= x0 D0 (x0
u) + ...
k m
= x0 D0 u + ...
(the points indicate the terms that have the order of differentiation with
respect to x0 less than m0 ). Hence
∗
τh,S
(a) = 0.
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3. If m ≥ m0 , then
lim ϕτh1 ,S (a)−m (x)[h1 (Y ϕm )(x) − Y (x)h1 (ϕm )(x)]
x→b,x∈S
/
=
=
lim ϕ−m (x)[h(Y ϕm+τh,S (a) )(x) − Y (x)h(ϕm+τh,S (a) )(x)]
x→b,x∈S
/
lim ϕτh,S (a)−(m+τh,S (a)) (x)
x→b,x∈S
/
× [h(Y ϕm+τh,S (a) )(x) − Y (x)h(ϕm+τh,S (a) )(x)]
= 0, by hypothesis.
Finally, we have
ϕτh1 ,S (a)−λ (x)h1 (ϕλ )(x) = ϕ−λ (x)h(ϕλ+τh,S (a) )(x)
= ϕτh,S (a)−(λ+τh,S (a)) (x)h(ϕλ+τh,S (a) )(x)
which tends to C(λ + τh,S (a), y) as x tends to y and x ∈
/ S.
This concludes the proof of the Theorem.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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5.
Formal Problem
If we choose a local card for which ϕ(x) = x0 and a = (0, . . . , 0) the Cauchy
problem (4.1) becomes (5.1) below. We devote this section to formal calculations by looking for solutions as power series of the problem (5.1) below connected with a Fuchsian operator P (x; D) of order m and weight m − k with
respect to x0 at x0 = 0.
We decompose this operator in the following form
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
P (x; D) = Pm (x; D0 ) − Q(x; D),
where
Pm (x; D0 ) =
k
X
m−p
am−p (x0 )xk−p
,
0 D0
Mekki Terbeche
p=0
Q(x; D) = −
X
µ(α0 )
x0
0
D0α0 (aα0 ,α0 (x0 , x0 )Dxα0 ),
α0 <m,|α|≤m
with am = 1 and µ(α0 ) = [α0 + 1 − (m − k)]+ .
Theorem 5.1. If the coefficients of Pm (x; D0 ) and Q(x; D) are holomorphic
functions near the origin in Cn+1 , then the following conditions are equivalent
i) For all integers λ ≥ m − k 6= 0,
ii) For any holomorphic Cauchy data uj , 0 ≤ j ≤ m − k − 1, near the origin
in Cn and for each holomorphic function f near the origin in Cn+1 , there
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exists a unique holomorphic solution u near the origin in Cn+1 solving
Cauchy problem
(5.1)
P (x; D)u(x) = f (x)
D0j u(0, x0 ) = uj (x0 ), 0 ≤ j ≤ m − k − 1.
j
P
0 x0
Suppose that the solution u(x0 , x0 ) has the form ∞
j=0 uj (x ) j! .
The problem is to determine uj (x0 ) for all j ≥ 0. It is easy to check the
following statements:
If
∞
X
xj
0
u(x0 , x ) =
uj (x0 ) 0
j!
j=0
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and
v(x0 , x0 ) =
∞
X
µ
0 x0
vµ (x )
µ=0
µ!
,
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then
(5.2)
" j #
∞
X j
X
xj
u(x0 , x0 )v(x0 , x0 ) =
uj−p (x0 )vp (x0 ) 0
p
j!
j=0
(5.3)
(5.4)
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
D0p u(x0 , x0 ) =
xq0 D0p u(x0 , x0 ) =
∞
X
j=0
∞
X
j=0
p=0
uj+p (x0 )
xj0
j!
[Cq (j)uj+p−q (x0 )]
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and by convention uk = 0 for k < 0.
By using (5.2), (5.3), and (5.4), one can check easily that
∞
X
xj0
Pm (x; D0 )u(x0 , x ) =
[D(j, x )uj+m−k (x )] ,
j!
j=0
0
where D(j, x0 ) =
C(j, x0 ) as
Pk
j=0
0
0
am−q (x0 )Ck−p (j) which can be written in terms of
C(j + m − k, x0 )
,
Cm−k (j + m − k)
D(j, x0 ) =
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
and we have
(5.5)
j
∞ X
C(j + m − k, x0 )
x
0
Pm (x; D0 )u(x0 , x ) =
uj+m−k (x ) 0 .
Cm−k (j + m − k)
j!
j=0
0
Similarly, if aα (x0 , x0 ) =
0
ν
0 0
ν=0 aα (x ) ν! , then
(5.6) Q(x; D)u(x0 , x ) = −
∞
X
×
X
p=0
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
X
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Cµ(α0 ) (j)

j=0
j+α0 −µ(α0 ) Title Page
xν
P∞
α0 <m,|α|≤m
j + α0 − µ(α0 )
p
Mekki Terbeche

j
0  x0
p
0
aα (x )D uj+α0 −µ(α0 )−p (x )
.
j!
α0
x0
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Finally, if f (x0 , x0 ) =
∞
P
j=0
xj
fj (x0 ) j!0 , then by using (5.5), (5.6), and by identifying
the coefficients of P (x; D)u(x) = f (x) we get the following expression
C(j + m − k, x0 )
uj+m−k (x0 )
Cm−k (j + m − k)
j+α0 −µ(α0 ) X
=−
X
Cµ(α0 ) (j)
p=0
α0 <m,|α|≤m
j + α0 − µ(α0 )
p
apα (x0 )
0
× Dxα0 uj+α0 −µ(α0 )−p (x0 ) + fj (x0 )
for all j ∈ N.
Lemma 5.2. Let P (λ; x0 ) =
Pm
k=0
ak (x0 )λk , (am = 1), be a polynomial in λ
∼
with continuous coefficients on some neighborhood V of the origin in Cn .
If P (j; 0) 6= 0 for all j ∈ N, there exists a neighborhood V of the origin such
that P (j; x0 ) 6= 0 for all x0 ∈ V and all j ∈ N .
Proof. We have
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
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m−1
X
m
0
0 k
|P (λ; x )| ≥ |λ| − ak (x )λ .
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k=0
Let
M=
max
∼
|ak (x0 )|
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0≤k≤m−1,x0 ∈V
then
"
0
m
|P (λ; x )| ≥ |λ|
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m−1
M X
1
1−
.
m−k−1
|λ| k=0 |λ|
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If |λ| > 1, then
|P (λ; x0 )| > 1 −
M
.
|λ| − 1
If x0 ∈ Ṽ and |λ| ≥ 2M + 1, then
1
|P (λ; x0 )| > .
2
∼
In other words, if j is an integer such that j ≥ 2M + 1 and x0 ∈ V then
P (j; x0 ) 6= 0.
Now let j ∈ N such that 0 ≤ j < 2M +1. Since P (j; 0) 6= 0, then by continuity,
there is a neighborhood of the origin Vj such that P (j; x0 ) 6= 0 for all x0 ∈ Vj .
In conclusion we choose V = (∩0≤j<2M +1 Vj ) ∩ Ṽ , and we have
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
0
P (j; x ) 6= 0
for all x0 ∈ V and all j ∈ N, as required.
Corollary 5.3. There exists a neighborhood V of the origin such that C (j +
m − k, x0 ) 6= 0 for all x0 ∈ V and all j ∈ N, and the induction formula

Cm−k (j + m − k)  X
(5.7) uj+m−k (x0 ) = −
Cµ(α0 ) (j)
C(j + m − k, x0 )
α0 <m,|α|≤m

j+α0 −µ(α0 ) X
0
j + α0 − µ(α0 )
×
apα (x0 )Dxα0 uj+α0 −µ(α0 )−p (x0 ) + fj (x0 )
p
p=0
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yields for all x0 ∈ V and all j ∈ N.
Under the conditions of the Theorem 5.1, there exists a unique formal series
∞
X
xj0
uj (x )
u(x0 , x ) =
j!
j=0
0
0
solution of the problem (5.1) since all uj (x0 ) are uniquely determined by
(5.7).
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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6.
Proof of the Main Theorem
Let h be a differential operator of Fuchsian type in a with respect to S of weight
τh,S (a) and of order m. We want to solve (4.1) in some neighborhood of a.
Set u − v = w and the problem (4.1) becomes
(6.1)
h(w) = g,
w = O(ϕτh,S (a) ),
where g = f − h(v).
It follows from the second condition of (6.1) that there is a unique holomorphic function U in some neighborhood of a such that w = O(ϕτh,S (a) ) and
finding U is equivalent to finding w.
U verifies
h(U ϕτh,S (a) ) = g,
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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i.e. U satisfies the equation
h1 (U ) = g,
where h1 is a Fuchsian operator of weight zero in a relative to S (by Theorem
4.2).
If we choose a local card such that ϕ(x) = x0 and a = (0, ..., 0); in this local
card the equation becomes
P̃ (U ) = Q(U ) + g,
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where
P̃ =
m
X
am−p (x0 )xm−p
D0m−p , (am = 1),
0
p=0
Q = −
m−1
X
xα0 0 +1 D0α0 Bm−α0 ,
α0 =0
and
0
X
Bm−α0 =
aα (x)Dxα0 .
|α0 |≤m−α0
Let us denote by C(j + m − k, x0 ), (respectively C1 (λ, x0 )) the Fuchsian
polynomial characteristic of h, (respectively h1 ). It follows from Lemma 5.2
that if C(j, 0) 6= 0 for all j ∈ N, j ≥ m − k, then there is a neighborhood V of
the origin in Cn for which C(j, x0 ) 6= 0 for all x0 ∈ V and all j ∈ N, j ≥ m − k
.Hence P̃ is one to one on the set of holomorphic functions at the origin.
j
P
0 x0
If u(x0 , x0 ) = ∞
j=0 uj (x ) j! then
P̃ −1 u(x) =
∞
X
j=0
uj (x0 ) xj0
,
C1 (j, x0 ) j!
and the problem (6.1) is equivalent to the following problem
(6.2)
U = P̃ −1 ◦ Q (U ) + U0 ,
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
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where
U0 = P̃ −1 (g).
Page 23 of 31
As in [1], [4], [5], [7], and [8], a successive approximation method will be
used in the following
sense.
Let Up+1 = P̃ −1 ◦ Q (Up ) + U0 for p ≥ 0, and set Vp = Up+1 − Up . Then
Vp+1 = P̃ −1 ◦ Q (Vp ).
Let D = (x0 , x0 ) : (x0 , x0 ) ∈ Cn+1 , |x0 | ≤ R1 and |x1 | + · · · + |xn | ≤ 1r ,
then V0 is holomorphic in some open neighborhood in D, and there is a constant
denoted kV0 k such that
Vp (x) kV0 k
P
where t(x0 ) = ni=0 xi , ξ0 ≥
open interval (0, 1).
R
,
η0
1
1
·
,
1 − x0 ξ0 1 − st(x0 )
s≥
r
,
η0
and η0 is some given number in the
Lemma 6.1. There exists a constant K such that
(6.3)
Vp (x) kV0 k
Kp
xp0
1
·
·
0
mp
(s − s)
1 − x0 ξ0 1 − s0 t(x0 )
for all s0 > s.
Proof. Clearly (6.3) holds for p = 0. Suppose (6.3) holds for p, and let us prove
it for p + 1.
We have
m−1
X
P̃ −1 ◦ Q =
P̃ −1 ◦ (xα0 0 +1 D0α0 Bm−α0 ),
α0 =0
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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and we want to study the action of the operators Bm−α0 , xα0 0 +1 D0α0 , and P̃ −1 on
Vp .
1) We have
X
Bm−α0 (x; Dx0 ) =
0
aα0 ,α0 (x0 , x0 )Dxα0 .
|α0 |≤m−α0
For all α such that |α| ≤ m there is Mα for which aα (x) Mα
· 1 .
1−x0 R 1−rt(x0 )
Set
Cm−α0 (x; Dx0 ) =
1
1
·
1 − x0 R 1 − rt(x0 )
X
0
Mα0 ,α0 Dxα0 .
|α0 |≤m−α0
By Definition 3.3, Bm−α0 (x; Dx0 ) Cm−α0 (x; Dx0 ). Let σ be in the open
interval (s, s0 ), then
1
xp0
1
Kp
Bm−α0 (Vp )(x) kV0 k
·
·
·
mp
(σ − s)
1 − Rx0 1 − ξ0 x0 1 − rt(x0 )
X
1
α0
×
Mα0 ,α0 Dx0
1 − σt(x0 )
0
|α |≤m−α0
1
xp0
1
Kp
kV0 k
·
·
·
(σ − s)mp 1 − Rx0 1 − ξ0 x0 1 − rt(x0 )
0
X
σ |α | |α0 |!
×
Mα0 ,α0
.
[1 − σt(x0 )]|α0 |+1
0
|α |≤m−α0
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
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One can check easily that
1
1
1
·
1 − Rx0 1 − ξ0 x0
1 − ξR0
1
1 − η0
·
1
1 − ξ0 x0
·
1
.
1 − ξ0 x0
By using [8], we obtain the following majoration
(m − α0 )!
|α0 |!
0
[1 − σt(x0 )]|α |+1
[1 − σt(x0 )]m−α0 +1
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
hence
Kp
1
xp0
·
·
(σ − s)mp 1 − η0 1 − ξ0 x0
X
(m − α0 )!
1
×
·
1 − rt(x0 ) [1 − σt(x0 )]m−α0 +1 0
Mekki Terbeche
Bm−α0 (Vp )(x) kV0 k
0
Mα0 ,α0 σ |α | .
|α |≤m−α0
∼
Again by [8], there exists C m−α0 (a, b) such that
∼
Kp
1
xp0
Bm−α0 (Vp )(x) kV0 k
·
Rα C m−α0 (a, b)
(σ − s)mp 1 − η0 0
1 − ξ0 x0
1
1
(m − α0 )!
×
·
· 0
0
0
0
1 − s t(x ) 1 − rt(x ) (s − σ)m−α0
P
0
where Rα0 = |α0 |≤m−α0 Mα0 ,α0 b|α | .
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If we let a >
r
,
η0
then
1
1
1
1
·
·
.
0
0
0
1 − s t(x ) 1 − rt(x )
1 − η0 1 − s0 t(x0 )
Finally,
Bm−α0 (Vp )(x) kV0 k
Rα0
Kp
·
(σ − s)mp (1 − η0 )2
∼
×
xp0
1
C m−α0 (a, b)
·
·
.
0
m−α
0
(s − σ)
1 − s0 t(x0 ) 1 − ξ0 x0
2) A straightforward computation leads to the following majoration
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
∼
xα0 0 +1 D0α0 Bm−α0 (Vp )(x)
Kp
Rα0
C m−α0 (a, b)
kV0 k
·
·
(σ − s)mp (1 − η0 )2 (s0 − σ)m−α0
" ∞
#
X j−p−1
1
×
ξ0
Cα0 (j − 1)xj0
.
0 t(x0 )
1
−
s
j=p+1
Set
wp (x) = xα0 0 +1 D0α0 Bm−α0 (Vp )(x) =
∞
X
j=0
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ξ0j−p−1 wp,j (x0 )xj0 ,
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hence
wp,j (x0 ) kV0 k
Kp
Rα0
·
mp
(σ − s)
(1 − η0 )2
∼
C m−α (a, b)
1
× 0 0 m−α0 · ξ0j−p−1 Cα0 (j − 1) ·
.
(s − σ)
1 − s0 t(x0 )
If
Fp (x) = P̃
−1
(wp )(x) =
∞
X
Fp,j (x0 )xj0 ,
j=0
then
Fp,j (x0 ) =
wp,j (x0 )
.
C1 (j, x0 )
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
It follows from Corollary 3.4 that
p
K
(σ − s)mp
ξ0j−p−1
×
·
m−α
(j + 1) 0
Kp
kV0 k
(σ − s)mp
Fp,j (x0 ) kV0 k
∼
Rα0
C m−α (a, b)
· 0 0 m−α0
2
(1 − η0 ) (s − σ)
1
1
·
0
1 − rt(x ) 1 − s0 t(x0 )
·
ξ0j−p−1
1
R̃α0
· 0
,
m−α0 ·
m−α0 ·
1 − s0 t(x0 )
(s − σ)
(p + 1)
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for all j ≥ p + 1, where R̃α0 =
∼
Rα 0
C
(a, b),
(1−η0 )3 m−α0
hence
Kp
R̃α0
Fp (x) kV0 k
·
m−α
mp
(σ − s)
[(p + 1)(s0 − σ)] 0
" ∞
#
X j−p−1 j
1
×
ξ0
x0
1 − s0 t(x0 )
j=p+1
R̃α0
1
Kp
xp+1
0
kV0 k
·
·
.
m−α0 ·
mp
0
(σ − s)
1 − ξ0 x0 1 − s0 t(x0 )
[(p + 1)(s − σ)]
If we choose σ such that s < σ < s0 and s0 − σ =
majoration holds
s0 −s
p+1
then the following
(b − a)α0
1
1
m
·
≤
e
m−α
(σ − s)mp [(p + 1)(s0 − σ)] 0
(s0 − s)m(p+1)
Mekki Terbeche
Title Page
and consequently
Kp
xp+1
1
0
m
α0
Fp (x) kV0 k 0
·
R̃
e
(b
−
a)
·
·
.
α0
m(p+1)
(s − s)
1 − ξ0 x0 1 − s0 t(x0 )
Finally
Vp+1 (x) kV0 k ·
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Kp
em
(s0 − s)m(p+1)
" m−1
#
X
×
R̃α0 (b − a)α0
α0 =0
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xp+1
1
0
·
.
1 − ξ0 x0 1 − s0 t(x0 )
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Choosing then
K = em
m−1
X
R̃α0 (b − a)α0
α0 =0
and yields easily the lemma.
If we impose |ξ0 x0 | ≤ ρ0 < 1 and b(|x1 | + · · · + |xn |) ≤ ρ0 < 1 then
p
1
K |x0 |
|Vp (x)| ≤ kV0 k
0
m
(s − s)
(1 − ρ0 )2
for all p ∈ N and all s0 > s.
0
m
If |x0 | ≤ (s −s)
, where K 0 > K, then the series of general term Vp conK0
verges normally and the sequence of general term Up converges uniformly
to a holomorphic function U on some suitable choice of polydiscs centered
at the origin in Cn+1 .
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
Title Page
Since
P̃ (Up+1 ) = Q(Up ) + g,
then the limit U satisfies the equation
P̃ (U ) = Q(U ) + g
therefore h1 (U ) = g as desired.
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References
[1] M.S. BAOUENDI AND C. GOULAOUIC, Cauchy problems with characteristic initial hypersurface, Comm. on Pure and Appl. Math., 26 (1973),
455–475.
[2] L.C. EVANS, Partial Differential Equations, AMS, Providence, RI, USA,
1998.
[3] Y. HASEGAWA, On the initial value problems with data on a characteristic
hypersurface, J. Math. Kyoto Univ., 13(3) (1973), 579–593.
[4] Y. HASEGAWA, On the initial value problems with data on a double characteristic hypersurface, J. Math. Kyoto Univ., 11(2) (1971), 357–372.
[5] J. LERAY, Problème de Cauchy I, Bull. Soc. Math., France, 85 (1957), 389–
430.
[6] M. TERBECHE, A geometric formulation of Baouendi-Goulaouic and
Hasegawa theorems for holomorphic Fuchsian operators, Int. J. Applied
Mathematics, 5(4) (2001), 419–429.
[7] M. TERBECHE, Problème de Cauchy pour des opérateurs holomorphes
de type de Fuchs, Thèse de Doctorat 3ème Cycle, Université des Sciences
et Techniques de Lille-I, Villeneuve d’Ascq, France, 1980.
[8] C. WAGSCHAL, Une généralisation du problème de Goursat pour des
systèmes d’équations intégro-différentielles holomorphes ou partiellement
holomorphes, J. Math. Pures et Appl., 53 (1974), 99–132.
Necessary and Sufficient
Condition for Existence and
Uniqueness of the Solution of
Cauchy Problem for
Holomorphic Fuchsian
Operators
Mekki Terbeche
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