38 N.M. Tri
C
∞
solutions was considered in [10]. The paper is organized as follows. In §2 we introduce notations used in the paper and state some auxiliary lemmas. In §3 we state the main theorem
and prove a theorem on C
∞
regularity of the solutions. In §4 we establish the Gevrey analyticity of the solutions.
2. Some notations and lemmas
First let us define the function F
h 2k
x
1
, x
2
, y
1
, y
2
= 1
2πh − 1 x
1
− y
1 h−1
x
2k+1 1
−y
2k+1 1
2k+1
+ i x
2
− y
2
. For j = 1, . . . , h − 1 we have see [11]
M
j 2k
F
h 2k
= 1
2πh − j − 1 x
1
− y
1 h− j −1
x
2k+1 1
−y
2k+1 1
2k+1
+ i x
2
− y
2
and M
h 2k
F
h 2k
= δx − y , where x = x
1
, x
2
, y = y
1
, y
2
and δ· is the Dirac function. We will denote by a bounded domain in
R
2
with piece-wise smooth boundary. We now state a lemma on the representation formula. Its proof can be found in [10].
L
EMMA
4.1. Assume that u ∈ C
h
¯
then we have
1 ux =
Z
− 1
h
F
h 2k
x, yM
h 2k
uy d y
1
d y
2
+ Z
∂
h−1
X
j =0
− 1
j
M
j 2k
u M
h− j −1 2k
F
h 2k
x, y
n
1
+ i y
2k 1
n
2
ds . For reason of convenience we shall use the following Heaviside function
θ z =
1 if
z ≥ 0 , if
z 0 . We then have the following formula
d
m
dt
m
z
n
= n . . . n − m + 1θ n − m + 1z
n−m
. We will use ∂
α 1
, ∂
β 2
,
γ
∂
α,β
, ∂
α 1y
, ∂
β 2y
and
γ
∂
y α,β
instead of
∂
α
∂ x
α 1
,
∂
β
∂ x
β 2
, x
γ 1
∂
α+β
∂ x
α 1
∂ x
β 2
,
∂
α
∂ y
α 1
,
∂
β
∂ y
β 2
and y
γ 1
∂
α+β
∂ y
α 1
∂ y
β 2
, respectively. Throughout the paper we use the following notation n =
n if
n ≥ 1 , 1
if n ≤ 0 ,
and C
n
= C
n
if n ≥ 1 ,
1 if
n ≤ 0 . Furthermore all constants C
i
which appear in the paper are taken such that they are greater than 1. Put h2k + 1 = r
. For any integer r ≥ 0 let Ŵ
r
denote the set of pair of multi-indices α, β such that Ŵ
r
= Ŵ
1 r
∪ Ŵ
2 r
where Ŵ
1 r
= {α, β : α ≤ r ,
2α + β ≤ r }, Ŵ
2 r
= {α, β : α ≥ r , α + β ≤
r − r } .
On the Gevrey analyticity of solutions 39
For a pair α, β we denote by α, β
∗
the minimum of r such that α, β ∈ Ŵ
r
. For any nonnegative integer t we define the following sieves
4
t
= {α, β, γ : α + β ≤ t, 2kt ≥ γ ≥ α + 2k + 1β − t } ,
4
t
= {α, β, γ : α, β, γ ∈ 4
t
, γ = α + 2k + 1β − t } .
Later on we will use the following properties of 4
t
: 4
t
contains a finite number of elements less than 2kt + 1
3
elements. If α, β, γ ∈ 4
t
, β ≥ 1, γ ≥ 1, then α, β − 1, γ − 1 ∈ 4
t −1
. For every α, β, γ ∈ 4
t
, α , β
, γ ∈ 4
t
we can express
γ
∂
α,β γ
∂
α ,β
as a linear combination of
γ
′
∂
α
′
,β
′
, where α
′
, β
′
, γ
′
∈ 4
t +t
. For every α, β, γ ∈ 4
t
, α
1
, β
1
∈ Ŵ
r
and a nonnegative integer m we can rewrite
γ − m
∂
α+α
1
−m,β+β
1
as
γ − m
∂
α
2
,β
2
∂
α
3
1
∂
β
3
2
where α
2
, β
2
, γ − m ∈ 4
t
and α
3
, β
3
∈ Ŵ
r−m
see [12]. For any nonnegative integer r let us define the norm
|u, |
r
= max
α
1
,β
1
∈Ŵ
r
∂
α
1
1
∂
β
1
2
u, +
max
α1,β1∈Ŵr α1≥1,β1≥1
max
x ∈ ¯
∂
h 1
∂
α
1
1
∂
β
1
2
ux ,
where |w, | = P
α,β,γ ∈4
h−1
max
x ∈ ¯
|
γ
∂
α,β
w x|.
For any nonnegative integer l let H
l loc
denote the space of all u such that for any compact K
in we have P
α,β,γ ∈4
l
k
γ
∂
α,β
uk
L
2
K
∞ . We note the following properties of
H
l loc
:
H
l loc
⊂ H
l loc
where H
l loc
stands for the standard Sobolev spaces.
H
4k+2 loc
⊂ H
2 loc
⊂ C.
If u ∈ H
l loc
and α, β, γ ∈ 4
t
, t ≤ l then
γ
∂
α,β
u ∈ H
l−t loc
.
The following lemma is due to Grushin see [12] L
EMMA
4.2. Assume that u ∈ D
′
and M
h 2k
u ∈ H
l loc
then u ∈
H
l+h loc
.
Next we define a space generalizing the space of analytic functions see for example [7]. Let L
n
and ¯ L
n
be two sequences of positive numbers, satisfying the monotonicity condition
i n
L
i
L
n−i
≤ AL
n
i = 1, 2 . . . ; n = 1, 2 . . . , where A is a positive constant. A function Fx, v, defined for x = x
1
, x
2
and for v = v
1
, . . . , v
µ
in a µ-dimensional open set E , is said to belong to the class C{L
n−a
; | ¯ L
n−a
; E } a is an integer if and only if Fx, v is infinitely differentiable and to every pair of compact subsets
⊂ and E ⊂ E there corre-
spond constants A
1
and A
2
such that for x ∈ and v ∈ E
∂
j +k
Fx, v ∂
x
j
1
1
∂ x
j
2
2
∂v
k
1
1
. . . ∂v
k
µ
µ
≤ A
1
A
j +k 2
L
j −a
¯ L
k−a
j
1
+ j
2
= j, X
k
i
= k; j, k = 0, 1, 2 . . . .
We use the notation L
−i
= 1 i = 1, 2, . . .. If Fx, v = f x, we simply write f x ∈ C{L
n−a
; }. Note that C{n; }, C{n
s
; } is the space of all analytic functions s-Gevrey functions, respectively, in . Extensive treatments of non-quasi analytic functions in particular,
the Gevrey functions can be found in [13, 14]. Finally we would like to mention the following lemma of Friedman [7].
40 N.M. Tri
L
EMMA
4.3. There exists a constant C
1
such that if gz is a positive monotone decreasing function, defined in the interval 0 ≤ z ≤ 1 and satisfying
gz ≤ 1
8
12
k
g z
1 − 6
k
n +
C z
n−r −1
n ≥ r + 2, C 0 ,
then gz CC
1
z
n−r −1
.
3. The main theorem
