Rend. Sem. Mat. Univ. Pol. Torino
Vol. 60, 2 (2002)
P. Marcolongo∗
ON THE SOLVABILITY IN GEVREY CLASSES OF A LINEAR
OPERATOR IN TWO VARIABLES
Abstract. We show non solvability results in Gevrey spaces G s for a linear
partial differential operator with a single real characteristic of constant
multiplicity m, m ≥ 3, provided s > m/(m − 2) + δ, where δ > 0 depends
on the order of the degeneracy of a suitable lower order term. In particular,
δ → 0 as the order of the degeneracy tends to +∞.
1. Introduction
The main aim of the present paper is to study in detail the local solvability of a model
linear partial differential operator in two variables. We recall that although most of the
well known classical operators, appearing in the basic theory of PDEs and in Mathematical Phisics, are solvable, non solvable operators exist, as proved first by Lewy [9],
and often of a very simple form. The example of Lewy was generalized by Hörmander
[6], who proved a necessary condition for the local solvability of partial differential
operators, given by the following
T HEOREM 1. Let the linear partial differential operator P with coefficients in
C ∞ () be solvable in , in the sense that for every f ∈ C0∞ () we can find a solution u ∈ D0 () of Pu = f . Then, for every compact set K ⊂  there exist a positive
constant C and an integer M ≥ 0 such that
Z
X
X
sup |D α f (x)|
sup |D α t Pϕ(x)|
(1)
f (x)ϕ(x) d x ≤
|α|≤M x∈K
|α|≤M x∈K
for all f , ϕ ∈ C0∞ (K ).
If an operator P is non locally solvable in the previous sense it is natural to analyze
its behaviour in Gevrey spaces, being intermediate classes between the analytic and the
C ∞ functions. More precisely, from now on we will refer to the following functional
setting.
Let K be a compact subset of  and C a positive fixed constant. Let us consider the
subspace G0s (, K , C), 1 < s < +∞, of C0∞ (), given by all the functions f (x) with
∗ The author is thankful to Professor Luigi Rodino and Professor Petar Popivanov for useful discussions
and hints on the subject of this paper.
101
102
P. Marcolongo
support contained in K such that, for some R ≥ 0,
sup |∂ α f (x)| ≤ R C |α| (α!)s .
(2)
x∈K
G0s (, K , C) is a Banach space endowed with the norm:
k f ks,K , C := sup C −|α| (α!)−s sup |∂ α f (x)|
(3)
α
x∈K
or equivalently
(4)
k f ks,K , C :=
X
C −|α| (α!)−s k∂ α f k L p (K )
α
with p ≥ 1 fixed. From now on we consider (4) with p = 2.
S
D EFINITION 1. G0s () = K , C G0s (, K , C) where K and C run respectively
over the set of all the compact sets contained in  and over the set of the positive real
numbers.
Therefore it is natural to endow G0s () with the inductive limit topology:
ind lim G0s (, K , C).
(5)
K %
C%+∞
Similarly we define G s () as the projective limit topology of the spaces G s (, K , C)
of all the functions f ∈ C ∞ () for which the norm (3) of the restriction to K is finite.
The main result that we will handle in the following, concerning with the solvability
in a Gevrey frame of partial differential operators with G s coefficients, is the Gevrey
version of Theorem 1 proved by Corli [2].
T HEOREM 2. Let s be a fixed real number, 1 < s < +∞. Let the linear partial
differential operator P be s-solvable in , i.e. for all f ∈ G0s () there exists u in
Ds0 (), space of the s-ultradistributions in , solution of Pu = f . Then for every
compact subset K of , for every η > > 0, there exists a positive constant C such
that:
2
t
(6)
max | f (x)| ≤ C k f k
1
1 k Pfk
x∈K
s,K , η−
s,K , η−
for every f ∈ G0s (, K , η1 ).
The previous theorem will be used in the following manner: an operator P is not
s-locally solvable in x 0 if the associated transposed equation t P f = 0 admits a suitable
sequence of approximated solutions which make the right hand side of (6) arbitrarily
small and let the left hand side bounded from below.
103
On the solvability in Gevrey
Now let us analyze the problem we are interested in.
Let us consider the operator
(7)
P = Dxm1 − A(x 1 )Dxm−1
− B(x 1 )Dxm−2
Dx1
2
2
with m odd, m ≥ 3; A(x 1 ), B(x 1 ) are analytic functions of x 1 , defined in a neighborhood of x 1 = 0.
The operator P is weakly hyperbolic in x 1 with a single characteristic of constant mulm
tiplicity m and therefore it is always locally G s solvable for s < m−1
without any
restrictions on A(x 1) and B(x 1 ). This well known fact follows from the G s well posedness of the Cauchy problem for P or from more general results about the G s solvability
of linear PDEs with multiple characteristics (e.g., cf. Corollary 5.1.3 in MascarelloRodino [13], see also [1], [5], [17]). Next if = m(A(x 1)) vanishes of odd order at
x 1 = 0, changing its sign from - to +, the results of Corli [3] imply that P is not locally
m
. We will investigate the case of = m A(x 1) vanishing of
solvable in G s for s > m−1
even order at x 1 = 0.
Let us suppose that for some integer h > 0:
(8)
(9)
<e A(0) 6= 0,
=m A(x 1) =
cx 12h
+ o(x 12h )
for
x 1 → 0, c 6= 0.
Moreover for a fixed l > 0:
(10)
=m B(x 1 ) = d x 1l + o(x 1l )
for
x 1 → 0, d 6= 0.
Popivanov [14] ( see also Popivanov-Popov [15]) proved that P is not locally solvable,
in the C ∞ sense, if h is sufficiently large with respect to l.
Moreover, if the conditions (8), (9), hold, as a particular case of Theorem 3.7 in [10] or
Theorem 3.2 in [4] we obtain that the operator (7) is s-solvable for
(11)
s<
m
.
m−2
At this point a natural question arises: what can we say about the behaviour of P for
m
indexes s ≥ m−2
? We get that P is non s-solvable for
(12)
s>
m
+ θ (h),
m −2
with θ (h) → 0 for h → 0.
The proof of this result, developed in the next section, is based on Theorem 2, applied
to a function f λ (x) of the following form:
fλ (x) = v(x) · ei
m−2
1
m−1
λψ0 (x)+λ m ψ1 (x)+λ m ψ2 (x)+...+λ m ψm−1 (x)
.
We will choose the phase functions ψi (x) according to a standard proceeding (see [3]).
At this point, solving suitable transport equations, we are able to find an amplitude
104
P. Marcolongo
function v(x) such that the approximate solution f (x) of the homogeneous equation
λ = 0 makes the right hand side of (6) arbitrarily small for λ sufficiently large and
s satisfying (12), but leaves the left hand side greater than or equal to 1. We observe
that the previous estimates need also a suitable version for this case of the results of
Ivrii [8]. For more details we also address to [12].
tPf
2. The main result
T HEOREM 3. Let us suppose that A (respectively, B) satisfies (8), (9) (respectively,
(10)) and
2(2l + 2) if m = 3
(13)
h≥
2l + 2
if m ≥ 5.
Then P is non s-solvable at the origin for every s satisfying (12) where
θ (h) =
m(l + 1)
.
(m − 2)[(2m − 4)h − (m − 1)l − 1]
Proof. We will reason ab absurdo.
First let us observe that
t
(14)
P = −(−i )m ∂xm1 − (−i )m−1 A(x 1 )∂xm−1
− (−i )m−1 B(x 1 )∂xm−2
∂x1 +
2
2
− (−i )m−1 B 0 (x 1 )∂xm−2
.
2
Let us define
(15)
fλ (x) := e
m−2
1
m−1
i λψ0 (x)+λ m ψ1 (x)+λ m ψ2 (x)+...+λ m ψm−1 (x)
where x = (x 1 , x 2 ).
1st STEP: choice of the phase functions ψi (x).
The basic idea is to apply the operator t P to f λ (x) and choose the phase functions
ψ0 (x), ψ1 (x), ψ2 (x) in order to make equal to zero the higher powers of λ.
Following a standard approach for constructing formal asymptotic solutions (e.g., cf.
[2], see also [5], Chapter IV) we choose ψ0 = x 2 and obtain
t
P fλ (x) = lm−1 (x)λm−1 + o(λm−1 ) f λ (x), λ → +∞.
where the coefficient is given by
(16)
lm−1 := −
∂ψ1 (x)
∂ x1
m
− A(x 1 ).
Letting lm−1 = 0, we obtain m complex solutions and we choose
Z x
e 1 ) d x1,
A(x
(17)
ψ1 (x) =
0
105
On the solvability in Gevrey
1
1
e 1 ) = [ A(x 1)] m is determined in such a way that [ A(0)] m is real-valued.
where A(x
e 1 ), for a new constant c.
Note that properties (8) and (9) keep valid for A(x
Then
(18)
=m ψ1 (x) =
c
x 2h+1 + o(x 12h+1).
2h + 1 1
Repeating the previous arguments with this choice of ψ1 (x) finally we find:
Z
1
e
B(x 1 ) d x 1 + i x 22
(19)
ψ2 =
m
with e
B(x 1 ) satisfying again (10) for a new constant d. Therefore
d l+1
1
l+1
x
+ o(x 1 ) + x 22 .
(20)
=m ψ2 (x) =
m l+1 1
The other phase functions are determined recursively, according to a standard proceeding, cf. [2]. We need not precise information on them, but they vanish at x 1 = 0. We
then write
f λ (x) = ei8(x)
(21)
where
m−2 1
d l+1
c
2h+1
2h+1
m
=m 8(x) =
+ o(x 1 ) + λ
x
x +
2h + 1 1
m l+1 1
m−3
+ o(x 1l+1 ) + x 22 + λ m o(x 1).
m−1
λ m
(22)
2nd STEP: the action of t P on u(x) = f λ (x) v(x).
According to our choices we find:
t
Pu(x) = f λ (x)t Pλ v(x)
setting
(23)
t
Pλ := λ
m−1
m
Q 0 (x, D) +
m−1
X
j =1
λ− j Q j (x, D)
where:
P
- Q 0 (x, D) = (−1)m
i ci (x)Di + S(x) , where ci (x) are polynomial in x and S(x)
is an analytic function;
- Q j (x, D) are differential operators of order less than or equal to m, with analytic
coefficients.
106
P. Marcolongo
3rd STEP: minimum of =m 8(x).
We follow here the arguments in Popivanov [14].
Obviously
(24)
=m 8(0, 0) = 0.
We are interested in the solutions of the following equations:
m−1
m−2 1
∂=m 8(x)
= λ m (cx 12h + o(x 12h )) + λ m
(d x 1l + o(x 1l ))+
∂ x1
m
(25)
m−3
m o(1)
= 0,
m−2
m−3
∂=m 8(x)
= 2λ m x 2 + λ m o(x 1) = 0.
∂ x2
+λ
(26)
The second equation is solved by
1
x 2 = o(λ− m x 1 ), for x 1 → 0.
(27)
Let us try to solve the first equation
λ
m−1
m
cx 12h +
1
2
1 −1 l
dλ m x 1 + o(x 12h ) + λ− m o(x 1l ) + λ− m o(1) = 0
m
Let us begin by considering the case l odd, c > 0, d < 0, 2h ≥ 2l + 2.
Taking large λ, in this way we have to solve
(28)
cx 12h +
2
1 −1 l
dλ m x 1 = o(λ− m ), for λ → +∞.
m
Let us define
1 1
− m 2h−l
(29)
:= λ
→ 0 if λ → 0.
1
From the previous definition it follows immediately that λ− m = 2h−l ; thus, replacing
this quantity in (28) we obtain the following equation:
(30)
cx 12h +
1 2h−l l
d
x 1 = o( 2(2h−l) ), → 0.
m
Let us set:
1
d 2h−l
;
x 1 = (1 + y1 ) −
mc
(31)
then (30) transforms into
(1 + y1 )l {(1 + y1 )2h−l − 1} = o( 2h−2l ).
107
On the solvability in Gevrey
Now let us consider the corresponding function
g(y1 , ) = (1 + y1l ){(1 + y1 )2h−l − 1} + o( 2h−2l ).
(32)
Clearly
g(0, 0) = 0,
∂g
∂g
(0, 0) = 2h − l > 0,
(0, 0) = 0.
∂y1
∂
Then, by the Implicit Function Theorem there exists a function y1 = y1 () ∈ C 2 , such
that g(y1, ) = 0, y1 (0) = 0, y10 (0) = 0. This implies that
y1 () = o( 2 ) for → 0.
Thus
1
1
d 2h−l
d 2h−l
x 1 = (1 + o( )) −
= −
+ o( 3 ).
mc
mc
2
Therefore, considering (29) and (27) we conclude that a critical point is given by
1
3
d 2h−l − 1 1
−
x1 λ = −
λ m 2h−l + o λ m(2h−l) , λ → +∞
mc
2h−l+1 −
x 2λ = o λ m(2h−l) , λ → +∞.
(33)
(34)
Now, we may give a complete picture of the behaviour of =m 8(x) varying l and sign
of d:
1. l odd;
- d > 0: =m 8(x) assumes its minimum at the origin and we have
=m 8(0, 0) = 0;
- d < 0: =m 8(x) assumes its minimum value in (x 1 λ, x 2 λ ) and, setting,
(35)
l+1 m−1
2h+1
2h − l
d 2h−l d
−
C0 λ := −
λ m m(2h−l) ,
mc
m (l + 1)(2h + 1)
δ1
we have
(36)
e
=m 8(x 1λ, x 2λ ) = C(λ)
= C0 λδ1 + o(λδ1 ).
where, being 2h ≥ 2l + 2 and m ≥ 3, we immediately get:
(37)
C0 < 0,
0 < δ1 =
2h + 1
m−1
−
< 1.
m
m(2h − l)
2. l even;
- d · c < 0: the expression of the critical point (x 1 λ , x 2 λ) and the behaviour of
=m 8(x) are the same described before;
108
P. Marcolongo
- d · c > 0: if d, c > 0 =m 8(x) is an increasing function of x 1 ; otherwise
=m 8(x) is a decreasing function of x 1 .
We may limit our attention to the case l odd and d < 0 (the case l even, c > 0, d < 0
is treated in the same way. Note moreover that the change of variable x 20 = −x 2 gives
a change of sign for the constant d).
We want to apply Taylor’s formula in order to analyze the behaviour of =m 8(x) near
the point (x 1 λ, x 2 λ ). To this aim let us observe that:
∂ 2 =m 8(x 1 , x 2 )
∂ x 22
= 2λ
m−2
m x2
+λ
m−3
m o(x 1 ),
m−3
∂ 2 =m 8(x 1 , x 2 )
= λ m o(1),
∂ x2∂ x1
m−1
∂ 2 =m 8(x 1 , x 2 )
m {2hcx 2h−1 +
=
λ
1
∂ x 12
1
− m l−1
1
x1
m dlλ
+ o(x 12h−1 )+
2
1
+λ− m o(x 1l−1 ) + λ− m o(1)}.
Therefore
(38)
e + C 0 (x 1 − x 1 λ )2 λδ2 + (x 2 − x 2 λ )2 λ
=m 8(x) = C(λ)
0
+
2h+1
X
(x 1 − x 1 λ ) j λ
m−1 2h+1− j
m − m(2h−l)
m−2
m +
+ o((x 2 − x 2 λ)3 )λ
j =3
where
2h−1
d 2h−l
C00 := 2c(2h − l) −
>0
mc
and
0 < δ2 :=
2h − 1
m −1
−
< 1.
m
m(2h − l)
Now let us suppose that
|x 1 − x 1 λ| ≤ 1 x 1 λ
with
0 < 1 1.
Being:
2h+1
X
(x 1 − x 1 λ) j λ
m−1 2h+1− j
m − m(2h−l)
<
j =3
1
2
< (x 1 − x 1 λ)2 λδ2 {1 x 1 λλ m(2h−l) + (1 x 12 λλ m(2h−l) + . . .
2h−1
. . . + (1 x 1 λ)2h−1 λ m(2h−l) }
m−1
m ,
109
On the solvability in Gevrey
we get
C00 (x 1 − x 1 λ)2 λδ2 +
(39)
2h+1
X
(x 1 − x 1 λ) j λ
m−1 2h+1− j
m − m(2h−l)
=
j =3
= C00 (x 1 − x 1 λ)2 λδ2 {1 + O(1 )}.
Now let us define:
(40)
(
1 if
g1(x 1 ) :=
0 if
x 1 ∈ [1 − 21 , 1 + 21 ]
x 1 ∈ [1 − 1 , 1 + 1 ].
Let us suppose that:
- g1 (x 1 ) ∈ G0s (R);
- 0 ≤ g1 (x 1 ) ≤ 1, for every x 1 in R.
By definition, it follows immediately that
g1 (x 1 x 1−1
λ ) = 1 if
|x 1 − x 1 λ| ≤
1
x1 λ
2
and
supp g1 (x 1 x 1−1
λ ) ⊂ {x 1 ∈ R : |x 1 − x 1 λ | ≤ 1 x 1 λ }.
Moreover
supp ∂∂x1 g1 (x 1 x 1−1
λ ) ⊂ {x 1 : (1 − 1 )x 1 λ ≤ |x 1 − x 1 λ | ≤ (1 −
{x 1 : (1 +
1
2 )x 1 λ
1
2 )x 1 λ }
[
≤ |x 1 − x 1 λ| ≤ (1 + 1 )x 1 λ}.
Then on suppg1(x 1 x 1−1
λ ), and for |x 2 − x 2 λ | ≤ 1 we have
(41)
m−2
e + C00 λδ2 (x 1 − x 1,λ )2 [1 + O(1 )] + 1 |x 1 − x 1 λ|2 λ m .
=m 8(x) ≥ C(λ)
2
1
In an analogous way on supp ∂g
∂ x 1 and for |x 2 − x 2 λ | ≤ 1 we get
(42)
with
(43)
e + C000 λδ1
=m 8(x) ≥ C(λ)
C000
:=
12 C00
2
d 2h−l
>0
−
mc
and 0 < δ1 < 1, defined as before.
Moreover let us consider:
(44)
(
1
g2 (x 2 ) :=
0
satisfying the following conditions
if
if
|x 2 | ≤
|x 2 | ≥
1
4
1
2
110
P. Marcolongo
- g2 (x 2 ) ∈ G0s (R);
- 0 ≤ g2 (x 1 ) ≤ 1, for every x 1 in R.
Let us suppose that x 2 ∈ supp g2 . Then
|x 2 − x 2 λ| ≤
1
2
+ |x 2 λ|.
Considering that x 2 λ → 0 for λ → +∞, for λ sufficiently large we have
|x 2 − x 2 λ| ≤ 1 .
Moreover, by definition we have
supp ∂g∂2x(x2 2 ) ⊂ { 41 ≤ |x 2 | ≤
Then, on supp
∂g2 (x 2 )
∂ x2 ,
1
2 }.
for |x 1 − x 1 λ| ≤ 1 x 1 λ and for λ sufficiently large we obtain
(45)
4t h STEP: choice of v(x).
Let us define
e + 12 λ
=m 8(x) ≥ C(λ)
m−2
m .
χ(x 1 , x 2 ) := g1 (x 1 x 1−1
λ ) g2 (x 2 ).
(46)
Now let us observe that the Cauchy Kovalevsky Theorem assures us the existence of
functions vλ(k) defined by induction as solutions of the following transport equations:

(0)
Q 0 (x, D)vλ = 0



j


 Q 0 (x, D)v (k) = − P λ− m Q j (x, D)v (k− j ) , k ≥ 1
λ
λ
(47)
j
 (0)



vλ(k) = 1 on x 1 = x 1 λ

vλ = 0 on x 1 = x 1 λ for k ≥ 1
where in the sum j runs from 1 to min{k, m − 1}.
Let us consider:
Vλ(N)
(48)
:=
N
X
vλ(k) .
k=0
Let us define
(N)
(49)
u λ (x) := χ(x) Vλ
(x) ei8(x)
5t h STEP: the conclusion.
We want to apply the necessary condition of Corli [3] to u λ (x). We recall that we want
to contradict the estimate (6) for f = u λ , λ → +∞, i.e.
2
t
(50)
max |u λ (x)| ≤ C ku λ k
1 k Pu λ k
1
K
s,K , η−
s,K , η−
111
On the solvability in Gevrey
where K is a suitable small compact neighborhood of the origin.
In view of (46), we may limit our attention to
(51)
K := {(x 1 , x 2 ) : |x 1 − x 1 λ| ≤ 1 x 1 λ, |x 2 | ≤
1
2 }.
Let us start evaluating the left hand side.
Let us observe that
e
|u λ (x 1 λ, x 2 λ )| = e−=m 8(x1 λ ,x2 λ ) = e−C(λ) ,
(52)
therefore
(53)
max |u λ (x)|
K
2
e
≥ e−2C(λ) .
t
1 .
1 k Pu λ k
s,K , η−
s,K , η−
Now let us analyze ku λ k
Let us reason as in Ivrii [8]: there exist positive constants B and N such that, setting
λ = 4Be L N,
L ≥ 1,
the following estimate holds:
(54)
with ν =
X
(k)
kD α vλ k
1
s,K , η
|α|≤m
m−1
m .
≤ Be
ν
−Lk+4M N m
,
k = 0, 1, 2, . . .
Therefore
X
(N)
kD α Vλ
k
|α|≤m
1
s,K , η
ν
m
≤ 2Be4M N .
Since our choice of vλ(k) implies
t
Pλ Vλ(N) =
m−2
X m−2
X
h
N−m+2+ j
λ− m Q h vλ
,
h=0 j =h
from (54) it follows that
(N)
kt Pλ Vλ
(55)
k
1
s,K , η
≤ Ce−L N+4M N
ν
m
.
Now
ku λ k
1
s,K , η−
≤ kχk
1
s,K , η+
≤ c e4M N
ν
m
(N)
kVλ
k
1 ke
s,K , η
i8
k
1
s,K , η+
kei8 k
1 .
s,K , η+
Let us recall now a useful result about Gevrey norms ( Corli [2], [3], Marcolongo [11],
Mascarello-Rodino [13], Gramchev-Popivanov [5]):
112
P. Marcolongo
P ROPOSITION 1. Let 9 be an analytic function in a neighborhood  of the origin
of Rn . Let us fix s > 1. Then for every compact K ⊂  and for every η > 0 we get:
keiλ9(x) k
(56)
1
s,K , η
≤ Ceaλ+d(λη)
1
1
s +dη s−1
,
where C and d are positive constants and a := sup K (−=m 9(x)).
Applying the previous proposition we obtain that:
≤ Cea+dλ
kei8 k
1
s,K , η+
e
with a = −C(λ).
Thus
(57)
ku λ k
1
s,K , η−
≤ Ce4M N
1
1
1
s (η+) s +d(η+) s−1
1
1
1
ν
m −C 0 λδ1 +dλ s (η+) s +d(η+) s−1
.
Regarding t Pu λ we get
kt Pu λ k
1
s,K , η−
≤ Ckt PVλ(N) k
1
s,K , η
(58)
+ Cλ
m−1
m
X
kei8 k
1
s,K , η
(N)
kD α Vλ
|α|≤m
k
1
s,K , η
kei8 k
1.
s,K ∩supp 5χ, η
Since K ∩ supp 5 χ doesn’t contain a small neighborhood of x 2 = 0, again by Proposition 1, considering also (42), we obtain that:
kei8 k
1
s,K ∩supp 5χ, η
e
00 δ
≤ Ce−C(λ)−C0 λ 1 +dλ
1
1 1
s η s +dη s−1
.
Therefore
(59)
t
k Pu λ k
1
s,K , η−
≤ Ce
1
ν
1
e
s +dη s−1
4M N m −C(λ)+d(λη)
{e−L N + λ
m−1
00 δ
m e −C 0 λ 1 }.
Summing up
t
1
1 k Pu λ k
s,K , η−
s,K , η−
ku λ k
≤ Ce8M N
+ C2 λ
ν
e
m −2C(λ)
{C1 e−L N+dλ
1 1
1
s (η s +(η+) s )
1 1
1
m−1
00 δ
s s
s
m e −C 0 λ 1 +dλ (η +(η+) ) }.
where C1 and C2 are positive constants depending on d, η, . But, from the previous
arguments,
e
t
−2C(λ)
,
ku λ k
1 ≥e
1 k Pu λ k
s,K , η−
s,K , η−
113
On the solvability in Gevrey
therefore:
1 ≤ C1 e−L N+8M N
+ C2 λ
ν
1 1
1
m +dλ s (η s +(η+) s )
1
1 1
ν
m−1
00 δ
m
s s
s
m e 8M N −C 0 λ 1 +dλ (η +(η+) ) ,
∀N.
Clearly the first addend of the right hand side tends to zero when N → +∞.
Regarding the second addend of the right hand side, in order to let it to zero when
N → +∞, we must impose
ν
< δ1
m
1
< δ1 .
s
(60)
(61)
Recalling that
δ1 =
m−1
2h + 1
−
m
m(2h − l)
we get that (60) is satisfied if the hypothesis (13) is valid and (61) is equivalent to
m
require s > m−2
+ θ (h). This concludes the proof.
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AMS Subject Classification: 35A07.
Paola MARCOLONGO
Dipartimento di Matematica
Università di Torino
via Carlo Alberto, 10
10123 Torino, ITALIA
e-mail: marcolongo@dm.unito.it
Lavoro pervenuto in redazione il 03.05.2001 e, in forma definitiva, il 28.01.2002.