On Continuation of Gevrey Class Solutions of Linear Partial
Transcript Of On Continuation of Gevrey Class Solutions of Linear Partial
J. Math. Sci. Univ. Tokyo 4 (1997), 551–593.
On Continuation of Gevrey Class Solutions of Linear Partial Differential Equations
By Akira Kaneko1
Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary
Abstract. We give a sufficient condition for the removability of thin singularities of Gevrey class solutions of linear partial differential equations. In §1 we give a sufficient condition for the removability in the case of equations with constant coefficients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients.
§0. Introduction
In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear partial differential equations. Some of the results given here are easily derived from Grushin’s pioneering works on continuation of C∞ solutions and from the author’s former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be obvious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-differentiable functions and ultradistributions.
Here is a brief plan of the present article. The first two sections treat equations with constant coefficients. In §1 we give a sufficient condition for
1991 Mathematics Subject Classification. 35G05, 35B60, 35E20.
1Partially supported by GRANT-IN-AID FOR SCIENTIFIC RESEARCH No. 07404003.
551
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Akira Kaneko
the removability of thin singularities of the Gevrey class solutions. This is a translation of Grushin’s work except for small details. In §2 we discuss the necessity of the condition given in §1. This is to construct non-trivial solutions with irremovable thin singularities under the condition opposite to §1. We generalize the construction of Grushin who gave such a few examples in his work [G2]. As an example, the precise Gevrey index for the threshold of existence of solutions with thin singularity is determined for the Schro¨dinger equation. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients. This is a modification of the author’s work for the removability of thin singularities of real analytic solutions.
For a general survey on this subject, we refer to [Kn11] for results until 1992, and [Kn12], where a list of open problems is gathered. The present article treats some of them concerning Gevrey class solutions.
§1. Continuation of Gevrey class solutions to equations with constant coefficients
Let P (D) be a linear partial differential operator with constant coef-
ficients, where P (ζ) is a polynomial in n variables ζ = (ζ1, . . . , ζn) and D = (D1, . . . , Dn) with Dj = −i∂/∂xj, j = 1, . . . , n. We define the two spaces of Gevrey class functions of index s by
(1.1)
E(s)(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∀h > 0, ∃CK,h > 0 sup |Dαf (x)| ≤ CK,hh|α|α!s for ∀α},
x∈K
and
(1.2) E{s}(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∃h = h(K) > 0,
∃C = C(K) > 0, sup |Dαf (x)| ≤ Ch|α|α!s for ∀α}.
x∈K
The first space has a simple topology of Fr´echet space and is easier to treat, but it is a little less natural because when s = 1, this corresponds to the space of entire functions. The second space has a very complicated topological structure and does not allow the closed range theorem to hold.
Continuation of Gevrey Class Solutions
553
Hence for this space we cannot utilize fundamental theorems such as the global surjectivity on convex open sets of linear partial differential operators with constant coefficients or the Fundamental Principle of EhrenpreisPalamodov. But it is more natural because for s = 1 this corresponds to the space of real analytic functions which is localizable along the real axis.
We also set
(1.3)
E1+(Ω) := E(s)(Ω) = E{s}(Ω).
s>1
s>1
This is a very convenient space, still containing enough functions in non-
quasianalytic ultra-differentiable class. Following the usage of Komatsu, we shall denote in the sequel by E∗(Ω) either of the spaces E(s)(Ω), E{s}(Ω), E1+(Ω) when we can state something commonly to these spaces. Thus E∗ denotes either of these function classes. In the same time, this symbol will denote the corresponding sheaf (that is, the localization) on Rn. As usual we let D∗(Ω) denote the functions of class E∗ with compact support
contained in Ω (together with the obvious topology if the dual space, that is,
the space of ultradistributions of this class, is considered). For a general set L we let E∗(L) denote the functions of class E∗ defined on a neighborhood
of L, with the obvious identification in the sense of inductive limit with
respect to the neighborhoods. In general, we denote by EP∗ (Ω) the space of solutions in Ω of the equa-
tion P (D)u = 0 of class E∗. Let K ⊂ Ω denote a thin compact subset.
Here “thin” means that the interior is void. We assume that it is contained
in a hyperplane, say ν · x = 0. (This follows automatically for convex thin
set, as we mainly consider in the sequel.) We study the continuation of solutions of P (D)u = 0 in E∗(Ω \ K) to solutions in E∗(Ω).
Proposition 1.1. The canonical map induced by the canonical restriction from Ω to Ω \ K:
EP∗ (Ω) → EP∗ (Ω \ K)
is injective. In other words, there are no solutions of P (D)u = 0 with compact support.
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Actually, any element in the kernel of the above map would be a solution of P (D)u = 0 with compact support. But via the Fourier transform we would then obtain P (ζ)u = 0, where u is entire, whence u ≡ 0.
Thus the quotient space
(1.4)
EP∗ (Ω \ K)/EP∗ (Ω)
will represent the obstruction for continuation of solutions of this class to
K. The reason why we restrict K to thin sets is obvious from the non-
quasianalyticity of the function class under consideration: If K had a nonvoid interior, then choosing f ∈ D∗(Int K) \ P (D)D∗(Int K) and a solution u ∈ E∗(Rn) of P (D)u = f , u|Ω\K would present a non-trivial element of (1.4).
First we shall show that the obstruction space (1.4) depends only on K
and not on Ω. For this purpose we recall the notion of local cohomology groups with coefficients in the solution sheaf EP∗ of class E∗ of the equation P u = 0.
Proposition 1.2. We have the following isomorphism
(1.5)
EP∗ (Ω \ K)/EP∗ (Ω) =∼ HK1 (Ω, EP∗ ).
More generally, for any set L containing K in its interior, we have
(1.5bis)
EP∗ (L \ K)/EP∗ (L) =∼ HK1 (L, EP∗ ).
The quotient space in (1.5) or (1.5bis) is determined by K only and does not depend on the choice of the neighborhoods.
Proof. Recall the following fundamental exact sequence of local cohomology groups:
(1.6)
0 → ΓK (Ω, EP∗ ) → Γ (Ω, EP∗ ) → Γ (Ω \ K, EP∗ ) → HK1 (Ω, EP∗ ) → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → H2(Ω, EP∗ ) = 0.
Continuation of Gevrey Class Solutions
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Here we have ΓK(Ω, EP∗ ) = 0 by Proposition 1.1. (The fact H2(Ω, EP∗ ) = 0 follows from the resolution (1.7) as will be discussed below.) We shall show
that the mapping
H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ )
is always injective. Then we will obtain the isomorphism (1.5). Incidentally, we obtain the exact sequence
(1.7)
0 → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → 0.
Recall now the following exact sequence of sheaves
(1.8)
0 → EP∗ → E ∗ P→(D) E ∗ → 0.
Here the surjectivity in the last arrow, that is, the local solvability in this class, is an easy consequence of the existence of a fundamental solution for a single linear partial differential operator P with constant coefficients. Taking the fundamental exact sequence of global cohomology groups on an open set Ω, we obtain
(1.9)
0 → Γ (Ω, EP∗ ) → Γ (Ω, E∗) P−(→D) Γ (Ω, E∗) → H1(Ω, EP∗ ) → H1(Ω, E∗) = 0.
The fact H1(Ω, E∗) = 0 is obvious because the sheaf E∗ is fine. (From this H2(Ω, EP∗ ) = 0 also follows.) Thus it suffices to show that the natural mapping induced from the restriction
E∗(Ω)/P (D)E∗(Ω) → E∗(Ω \ K)/P (D)E∗(Ω \ K)
is injective. Suppose that u ∈ E∗(Ω) represents an element mapped to 0. This implies that there exists v ∈ E∗(Ω \ K) such that
u|Ω\K = P (D)v.
Employing partitions of unity, construct h ∈ E∗(Ω) and w ∈ E∗(Rn \ K) such that v = h − w on Ω \ K. We can obviously choose w in such a way
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Akira Kaneko
that w ≡ 0 outside a ball of some radius R > 0, by cutting v smoothly in the class E∗ on a neighborhood of K. Put
P (D)w on Rn \ K, g=
P (D)h − u on Ω.
This definition is consistent on the common domain:
P (D)w = P (D)h − P (D)v = P (D)h − u on Ω \ K.
Thus g becomes a well defined element of E∗(Rn), which has compact support by the choice of w as above. Let f ∈ E∗(Rn) be a solution of P (D)f = g. (We can simply obtain such f by convoluting g with the distributional fundamental solution of P (D) which obviously preserves the Gevrey regularity.) Then, on Ω we have
u = P (D)(h − f ).
Hence, it represents 0 in E∗(Ω)/P (D)E∗(Ω). The proof for general neighborhood L of K is just similar. Now that
the isomorphism (1.5) is established, the final conclusion follows from the excision theorem of local cohomology groups.
Remark. 1) We cannot expect H1(Ω, EP∗ ) = 0, unless we have the global surjectivity of P (D) on Ω in this function class. This follows from the exact sequence (1.8). To have this surjectivity for open Ω, we first of all need to assume that Ω is convex. Then it is valid for the class E(s) (see e.g. Bjo¨rck [Bj1]), but still not in general for E{s} (see e.g. Cattabriga [C1]). The above method of argument was first introduced by [Kn6] for the real analytic solutions, to which the global surjectivity is neither available. Note that the above Proposition (or the sequence (1.7)) implies that in such a situation, the obstruction for the global surjectivity is concentrated on the neighborhood of ∂Ω.
2) We have an alternative choice of neighborhoods of K for which the global surjectivity holds. It is to take compact neighborhoods L ⊃⊃ K. In this case the surjectivity of P (D) : E∗(L) → E∗(L) holds irrespective of the convexity of L, because of the non-quasianalytic property of our class. This
Continuation of Gevrey Class Solutions
557
fact was implicitly employed by some of the proofs for the corresponding assertions in the author’s former publications.
In view of the above Proposition, we can always assume Ω to be convex, thus allowing to apply the Fourier analysis. Sometimes the choice of convex compact L simplifies the situation further.
Next, we shall show that the obstruction (1.4) can be decomposed via the irreducible components of P (D):
Corollary 1.3. Let Q be any factor of P . Then we have a canonical injection
(1.10a)
H
1 K
(Ω
,
EQ∗
)
→
H
1 K
(Ω
,
EP∗
).
Conversely, let P = Qm1 1 · · · QmNN be the decomposition of P (ζ) into different irreducible components with their multiplicities counted. Then we have a
(non-canonical ) injection
(1.10b)
N
H
1 K
(Ω
,
EP∗
)
→
[HK1 (Ω, EQ∗ j )]mj .
j=1
Hence we have HK1 (Ω, EP∗ ) = 0 if and only if HK1 (Ω, EQ∗ j ) = 0 for j = 1, . . . , N .
Proof. Let P = QR be a decomposition of polynomial. (We do not assume that Q, R are mutually prime.) Note that we have the following exact sequence of sheaves similar to (1.8):
(1.11)
0 → EQ∗ → EP∗ Q→(D) ER∗ → 0.
As a matter of fact, the exactness is obvious except for the surjectivity of the last arrow. But any solution u ∈ E∗ of Q(D)u = f for f ∈ ER∗ will satisfy P (D)u = R(D)(Q(D)u) = R(D)f = 0. Taking the fundamental
exact sequence of the relative cohomology groups for an open neighborhood
Ω ⊃ K, we obtain from (1.11) the following exact sequence:
(1.12)
0 → ΓK (Ω, EQ∗ ) → ΓK (Ω, EP∗ ) → ΓK (Ω, ER∗ )
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Akira Kaneko
→ HK1 (Ω, EQ∗ ) → HK1 (Ω, EP∗ ) → HK1 (Ω, ER∗ ) → HK2 (Ω, EQ∗ ) → HK2 (Ω, EP∗ ).
Here the terms in the first row vanish because of the absence of solutions with compact support. Hence the existence of canonical inclusion mapping (1.10a) follows. Since E∗ is not flabby, the second degree relative cohomology groups do not vanish even for single equations. But we have at least the injection mapping
H
1 K
(Ω
,
EP∗
)/H
1 K
(Ω
,
EQ∗
)
→
H
1 K
(Ω
,
ER∗
),
whence in view of the complete reducibility of the vector spaces, we have a (non-canonical) injection mapping
(1.13)
H
1 K
(Ω
,
EP∗
)
→
H
1 K
(Ω
,
EQ∗
)
⊕
H
1 K
(Ω
,
ER∗
).
Repeating this argument for Q, R, we finally obtain an injection like (1.10a).
Remark. In the preprint version of this article, we gave a proof for the
assertion that (1.10b) is an algebraic isomorphism, which was wrong as the
referee kindly pointed out. Here we give another proof of the isomorphism
for curiosity’s sake, although it will not be useful because we cannot give a
canonical mapping.
To prove an abstract isomorphism, it suffices to show that both sides of
(1.10b) have algebraic dimension (over C always) of the same cardinality.
Note that in view of Corollary 1.3 the algebraic dimension of each side of
(1.10b) is estimated by a finite multiple of the other’s. Thus it suffices to
show
that
each
H
1 K
(Ω
,
EQ∗
)
is
either
0
or
is
infinite
dimensional.
Suppose
that it has a non-zero finite dimension, and let u ∈ EQ∗ (Ω \ K) represent
a non-trivial element. Choose R which is irreducible and not contained
in the factors of Q. Then R(D)ju, j = 0, 1, 2, . . . will define elements of
HK1 (Ω, EQ∗ ) of which a finite number are linearly dependent, say
m
S(D)u := cjR(D)ju = v,
j=0
Continuation of Gevrey Class Solutions
559
v ∈ EP∗ (Ω), cj ∈ C , j = 0, 1, 2, . . . , m, cm = 0.
The simultaneous equation
S(D)w = v,
Q(D)w = 0
has a solution w ∈ E∗(Ω) as long as we shrink Ω a little for the fear of the
case of ∗ = {s} type space. (We neglect to introduce a new notation for the shrinked domain.) Then u − w ∈ E∗(Ω \ K) will satisfy
Q(D)(u − w) = 0,
S(D)(u − w) = 0.
Obviously, Q and S are primary to each other. Hence they define an overde-
termined system, and by Ehrenpreis-Malgrange’s classical theorem the so-
lution u − w can be continued to K. (Though the theorem may not have been written down for the class E∗, it is easy to modify their theory to
this case. A more easy-going way is that if ever we have a continuation as a C∞-solution, we can show that it is in class E∗ via the propagation of E∗ regularity for solutions of, say, Q(D)u = 0 up to K. This propagation
theorem can be shown by a standard argument employing a cut-off func-
tion in this class and a distribution fundamental solution of Q(D) by which the convolution preserves the E∗ regularity.) Thus u − w, hence u, can be
continued to a solution of Q(D)u = 0 near K, and irrespective of the fact of shrinking Ω, we conclude that u ∈ EQ∗ (Ω) for the original Ω, which is a contradiction.
Here we recall the notion of irregularity of a characteristic direction. We adopt the following definition. Let P (ζ) be an irreducible polynomial of order m such that Pm(ν) = 0, where Pm denotes the principal part. Consider
Q(s, t) := P (tξ + sν) = q0(ξ)tm + q1(s; ξ)tm−1 + · · · + qm(s; ξ)
as a polynomial of the two variables (s, t). For generically fixed ξ, let κ be the minimum value of the leading powers of the Puiseux expansions of the roots of Q for t in terms of s representing irreducible germs of N (Q) passing through the point (∞, 0) at infinity. Then we set µ = (1 − κ)−1
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Akira Kaneko
and call it the multiplicity of ν. A more exact definition may be given via the Newton polygon, transforming the point at infinity to the origin: Set σ = 1/s, τ = t/s and let
q(σ, τ ) = s−mQ(s, t) s=1/σ,t=τ/σ.
Factorize it as a polynomial of τ with coefficients in Oσ,0 at 0. Then µ is the inverse of the minimum value with respect to the irreducible factors of the leading powers of the Puiseux expansions of the roots of them.
Notice that the irregularity employed here is the mildest one, in comparison with the strongest one which is usually used e.g. to define the hyperbolicity.
Our main result here is the following
Theorem 1.4. Let K be a compact set contained in a hyperplane ν · x = 0. Assume further that every irreducible component of P (ζ) has ν as characteristic direction of irregularity ≤ µ. Then
EP(s)(Ω \ K)/EP(s)(Ω) = 0, if s ≤ µ/(µ − 1),
EP{s}(Ω \ K)/EP{s}(Ω) = 0, if s < µ/(µ − 1).
Corollary 1.5. Assume that every irreducible component of P (ζ) is non-elliptic. Then the isolated singularities of solutions of class E1+ are
removable, that is,
(1.14)
EP1+(Ω \ {0})/EP1+(Ω) = 0.
Remark that this sufficient condition on P (D) is the same as the one for the removablilty of isolated singularities of real analytic solutions given in [Kn1]–[Kn2].
Although the proof of Theorem 1.4 is almost a literal translation of Grushin’s original article [G2] for the removability of isolated singularities of C∞ solutions, we shall reproduce it here in detail, because it is nevertheless
On Continuation of Gevrey Class Solutions of Linear Partial Differential Equations
By Akira Kaneko1
Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary
Abstract. We give a sufficient condition for the removability of thin singularities of Gevrey class solutions of linear partial differential equations. In §1 we give a sufficient condition for the removability in the case of equations with constant coefficients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients.
§0. Introduction
In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear partial differential equations. Some of the results given here are easily derived from Grushin’s pioneering works on continuation of C∞ solutions and from the author’s former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be obvious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-differentiable functions and ultradistributions.
Here is a brief plan of the present article. The first two sections treat equations with constant coefficients. In §1 we give a sufficient condition for
1991 Mathematics Subject Classification. 35G05, 35B60, 35E20.
1Partially supported by GRANT-IN-AID FOR SCIENTIFIC RESEARCH No. 07404003.
551
552
Akira Kaneko
the removability of thin singularities of the Gevrey class solutions. This is a translation of Grushin’s work except for small details. In §2 we discuss the necessity of the condition given in §1. This is to construct non-trivial solutions with irremovable thin singularities under the condition opposite to §1. We generalize the construction of Grushin who gave such a few examples in his work [G2]. As an example, the precise Gevrey index for the threshold of existence of solutions with thin singularity is determined for the Schro¨dinger equation. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients. This is a modification of the author’s work for the removability of thin singularities of real analytic solutions.
For a general survey on this subject, we refer to [Kn11] for results until 1992, and [Kn12], where a list of open problems is gathered. The present article treats some of them concerning Gevrey class solutions.
§1. Continuation of Gevrey class solutions to equations with constant coefficients
Let P (D) be a linear partial differential operator with constant coef-
ficients, where P (ζ) is a polynomial in n variables ζ = (ζ1, . . . , ζn) and D = (D1, . . . , Dn) with Dj = −i∂/∂xj, j = 1, . . . , n. We define the two spaces of Gevrey class functions of index s by
(1.1)
E(s)(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∀h > 0, ∃CK,h > 0 sup |Dαf (x)| ≤ CK,hh|α|α!s for ∀α},
x∈K
and
(1.2) E{s}(Ω) := {f (x) ∈ C∞(Ω); ∀K ⊂⊂ Ω, ∃h = h(K) > 0,
∃C = C(K) > 0, sup |Dαf (x)| ≤ Ch|α|α!s for ∀α}.
x∈K
The first space has a simple topology of Fr´echet space and is easier to treat, but it is a little less natural because when s = 1, this corresponds to the space of entire functions. The second space has a very complicated topological structure and does not allow the closed range theorem to hold.
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553
Hence for this space we cannot utilize fundamental theorems such as the global surjectivity on convex open sets of linear partial differential operators with constant coefficients or the Fundamental Principle of EhrenpreisPalamodov. But it is more natural because for s = 1 this corresponds to the space of real analytic functions which is localizable along the real axis.
We also set
(1.3)
E1+(Ω) := E(s)(Ω) = E{s}(Ω).
s>1
s>1
This is a very convenient space, still containing enough functions in non-
quasianalytic ultra-differentiable class. Following the usage of Komatsu, we shall denote in the sequel by E∗(Ω) either of the spaces E(s)(Ω), E{s}(Ω), E1+(Ω) when we can state something commonly to these spaces. Thus E∗ denotes either of these function classes. In the same time, this symbol will denote the corresponding sheaf (that is, the localization) on Rn. As usual we let D∗(Ω) denote the functions of class E∗ with compact support
contained in Ω (together with the obvious topology if the dual space, that is,
the space of ultradistributions of this class, is considered). For a general set L we let E∗(L) denote the functions of class E∗ defined on a neighborhood
of L, with the obvious identification in the sense of inductive limit with
respect to the neighborhoods. In general, we denote by EP∗ (Ω) the space of solutions in Ω of the equa-
tion P (D)u = 0 of class E∗. Let K ⊂ Ω denote a thin compact subset.
Here “thin” means that the interior is void. We assume that it is contained
in a hyperplane, say ν · x = 0. (This follows automatically for convex thin
set, as we mainly consider in the sequel.) We study the continuation of solutions of P (D)u = 0 in E∗(Ω \ K) to solutions in E∗(Ω).
Proposition 1.1. The canonical map induced by the canonical restriction from Ω to Ω \ K:
EP∗ (Ω) → EP∗ (Ω \ K)
is injective. In other words, there are no solutions of P (D)u = 0 with compact support.
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Akira Kaneko
Actually, any element in the kernel of the above map would be a solution of P (D)u = 0 with compact support. But via the Fourier transform we would then obtain P (ζ)u = 0, where u is entire, whence u ≡ 0.
Thus the quotient space
(1.4)
EP∗ (Ω \ K)/EP∗ (Ω)
will represent the obstruction for continuation of solutions of this class to
K. The reason why we restrict K to thin sets is obvious from the non-
quasianalyticity of the function class under consideration: If K had a nonvoid interior, then choosing f ∈ D∗(Int K) \ P (D)D∗(Int K) and a solution u ∈ E∗(Rn) of P (D)u = f , u|Ω\K would present a non-trivial element of (1.4).
First we shall show that the obstruction space (1.4) depends only on K
and not on Ω. For this purpose we recall the notion of local cohomology groups with coefficients in the solution sheaf EP∗ of class E∗ of the equation P u = 0.
Proposition 1.2. We have the following isomorphism
(1.5)
EP∗ (Ω \ K)/EP∗ (Ω) =∼ HK1 (Ω, EP∗ ).
More generally, for any set L containing K in its interior, we have
(1.5bis)
EP∗ (L \ K)/EP∗ (L) =∼ HK1 (L, EP∗ ).
The quotient space in (1.5) or (1.5bis) is determined by K only and does not depend on the choice of the neighborhoods.
Proof. Recall the following fundamental exact sequence of local cohomology groups:
(1.6)
0 → ΓK (Ω, EP∗ ) → Γ (Ω, EP∗ ) → Γ (Ω \ K, EP∗ ) → HK1 (Ω, EP∗ ) → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → H2(Ω, EP∗ ) = 0.
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555
Here we have ΓK(Ω, EP∗ ) = 0 by Proposition 1.1. (The fact H2(Ω, EP∗ ) = 0 follows from the resolution (1.7) as will be discussed below.) We shall show
that the mapping
H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ )
is always injective. Then we will obtain the isomorphism (1.5). Incidentally, we obtain the exact sequence
(1.7)
0 → H1(Ω, EP∗ ) → H1(Ω \ K, EP∗ ) → HK2 (Ω, EP∗ ) → 0.
Recall now the following exact sequence of sheaves
(1.8)
0 → EP∗ → E ∗ P→(D) E ∗ → 0.
Here the surjectivity in the last arrow, that is, the local solvability in this class, is an easy consequence of the existence of a fundamental solution for a single linear partial differential operator P with constant coefficients. Taking the fundamental exact sequence of global cohomology groups on an open set Ω, we obtain
(1.9)
0 → Γ (Ω, EP∗ ) → Γ (Ω, E∗) P−(→D) Γ (Ω, E∗) → H1(Ω, EP∗ ) → H1(Ω, E∗) = 0.
The fact H1(Ω, E∗) = 0 is obvious because the sheaf E∗ is fine. (From this H2(Ω, EP∗ ) = 0 also follows.) Thus it suffices to show that the natural mapping induced from the restriction
E∗(Ω)/P (D)E∗(Ω) → E∗(Ω \ K)/P (D)E∗(Ω \ K)
is injective. Suppose that u ∈ E∗(Ω) represents an element mapped to 0. This implies that there exists v ∈ E∗(Ω \ K) such that
u|Ω\K = P (D)v.
Employing partitions of unity, construct h ∈ E∗(Ω) and w ∈ E∗(Rn \ K) such that v = h − w on Ω \ K. We can obviously choose w in such a way
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Akira Kaneko
that w ≡ 0 outside a ball of some radius R > 0, by cutting v smoothly in the class E∗ on a neighborhood of K. Put
P (D)w on Rn \ K, g=
P (D)h − u on Ω.
This definition is consistent on the common domain:
P (D)w = P (D)h − P (D)v = P (D)h − u on Ω \ K.
Thus g becomes a well defined element of E∗(Rn), which has compact support by the choice of w as above. Let f ∈ E∗(Rn) be a solution of P (D)f = g. (We can simply obtain such f by convoluting g with the distributional fundamental solution of P (D) which obviously preserves the Gevrey regularity.) Then, on Ω we have
u = P (D)(h − f ).
Hence, it represents 0 in E∗(Ω)/P (D)E∗(Ω). The proof for general neighborhood L of K is just similar. Now that
the isomorphism (1.5) is established, the final conclusion follows from the excision theorem of local cohomology groups.
Remark. 1) We cannot expect H1(Ω, EP∗ ) = 0, unless we have the global surjectivity of P (D) on Ω in this function class. This follows from the exact sequence (1.8). To have this surjectivity for open Ω, we first of all need to assume that Ω is convex. Then it is valid for the class E(s) (see e.g. Bjo¨rck [Bj1]), but still not in general for E{s} (see e.g. Cattabriga [C1]). The above method of argument was first introduced by [Kn6] for the real analytic solutions, to which the global surjectivity is neither available. Note that the above Proposition (or the sequence (1.7)) implies that in such a situation, the obstruction for the global surjectivity is concentrated on the neighborhood of ∂Ω.
2) We have an alternative choice of neighborhoods of K for which the global surjectivity holds. It is to take compact neighborhoods L ⊃⊃ K. In this case the surjectivity of P (D) : E∗(L) → E∗(L) holds irrespective of the convexity of L, because of the non-quasianalytic property of our class. This
Continuation of Gevrey Class Solutions
557
fact was implicitly employed by some of the proofs for the corresponding assertions in the author’s former publications.
In view of the above Proposition, we can always assume Ω to be convex, thus allowing to apply the Fourier analysis. Sometimes the choice of convex compact L simplifies the situation further.
Next, we shall show that the obstruction (1.4) can be decomposed via the irreducible components of P (D):
Corollary 1.3. Let Q be any factor of P . Then we have a canonical injection
(1.10a)
H
1 K
(Ω
,
EQ∗
)
→
H
1 K
(Ω
,
EP∗
).
Conversely, let P = Qm1 1 · · · QmNN be the decomposition of P (ζ) into different irreducible components with their multiplicities counted. Then we have a
(non-canonical ) injection
(1.10b)
N
H
1 K
(Ω
,
EP∗
)
→
[HK1 (Ω, EQ∗ j )]mj .
j=1
Hence we have HK1 (Ω, EP∗ ) = 0 if and only if HK1 (Ω, EQ∗ j ) = 0 for j = 1, . . . , N .
Proof. Let P = QR be a decomposition of polynomial. (We do not assume that Q, R are mutually prime.) Note that we have the following exact sequence of sheaves similar to (1.8):
(1.11)
0 → EQ∗ → EP∗ Q→(D) ER∗ → 0.
As a matter of fact, the exactness is obvious except for the surjectivity of the last arrow. But any solution u ∈ E∗ of Q(D)u = f for f ∈ ER∗ will satisfy P (D)u = R(D)(Q(D)u) = R(D)f = 0. Taking the fundamental
exact sequence of the relative cohomology groups for an open neighborhood
Ω ⊃ K, we obtain from (1.11) the following exact sequence:
(1.12)
0 → ΓK (Ω, EQ∗ ) → ΓK (Ω, EP∗ ) → ΓK (Ω, ER∗ )
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Akira Kaneko
→ HK1 (Ω, EQ∗ ) → HK1 (Ω, EP∗ ) → HK1 (Ω, ER∗ ) → HK2 (Ω, EQ∗ ) → HK2 (Ω, EP∗ ).
Here the terms in the first row vanish because of the absence of solutions with compact support. Hence the existence of canonical inclusion mapping (1.10a) follows. Since E∗ is not flabby, the second degree relative cohomology groups do not vanish even for single equations. But we have at least the injection mapping
H
1 K
(Ω
,
EP∗
)/H
1 K
(Ω
,
EQ∗
)
→
H
1 K
(Ω
,
ER∗
),
whence in view of the complete reducibility of the vector spaces, we have a (non-canonical) injection mapping
(1.13)
H
1 K
(Ω
,
EP∗
)
→
H
1 K
(Ω
,
EQ∗
)
⊕
H
1 K
(Ω
,
ER∗
).
Repeating this argument for Q, R, we finally obtain an injection like (1.10a).
Remark. In the preprint version of this article, we gave a proof for the
assertion that (1.10b) is an algebraic isomorphism, which was wrong as the
referee kindly pointed out. Here we give another proof of the isomorphism
for curiosity’s sake, although it will not be useful because we cannot give a
canonical mapping.
To prove an abstract isomorphism, it suffices to show that both sides of
(1.10b) have algebraic dimension (over C always) of the same cardinality.
Note that in view of Corollary 1.3 the algebraic dimension of each side of
(1.10b) is estimated by a finite multiple of the other’s. Thus it suffices to
show
that
each
H
1 K
(Ω
,
EQ∗
)
is
either
0
or
is
infinite
dimensional.
Suppose
that it has a non-zero finite dimension, and let u ∈ EQ∗ (Ω \ K) represent
a non-trivial element. Choose R which is irreducible and not contained
in the factors of Q. Then R(D)ju, j = 0, 1, 2, . . . will define elements of
HK1 (Ω, EQ∗ ) of which a finite number are linearly dependent, say
m
S(D)u := cjR(D)ju = v,
j=0
Continuation of Gevrey Class Solutions
559
v ∈ EP∗ (Ω), cj ∈ C , j = 0, 1, 2, . . . , m, cm = 0.
The simultaneous equation
S(D)w = v,
Q(D)w = 0
has a solution w ∈ E∗(Ω) as long as we shrink Ω a little for the fear of the
case of ∗ = {s} type space. (We neglect to introduce a new notation for the shrinked domain.) Then u − w ∈ E∗(Ω \ K) will satisfy
Q(D)(u − w) = 0,
S(D)(u − w) = 0.
Obviously, Q and S are primary to each other. Hence they define an overde-
termined system, and by Ehrenpreis-Malgrange’s classical theorem the so-
lution u − w can be continued to K. (Though the theorem may not have been written down for the class E∗, it is easy to modify their theory to
this case. A more easy-going way is that if ever we have a continuation as a C∞-solution, we can show that it is in class E∗ via the propagation of E∗ regularity for solutions of, say, Q(D)u = 0 up to K. This propagation
theorem can be shown by a standard argument employing a cut-off func-
tion in this class and a distribution fundamental solution of Q(D) by which the convolution preserves the E∗ regularity.) Thus u − w, hence u, can be
continued to a solution of Q(D)u = 0 near K, and irrespective of the fact of shrinking Ω, we conclude that u ∈ EQ∗ (Ω) for the original Ω, which is a contradiction.
Here we recall the notion of irregularity of a characteristic direction. We adopt the following definition. Let P (ζ) be an irreducible polynomial of order m such that Pm(ν) = 0, where Pm denotes the principal part. Consider
Q(s, t) := P (tξ + sν) = q0(ξ)tm + q1(s; ξ)tm−1 + · · · + qm(s; ξ)
as a polynomial of the two variables (s, t). For generically fixed ξ, let κ be the minimum value of the leading powers of the Puiseux expansions of the roots of Q for t in terms of s representing irreducible germs of N (Q) passing through the point (∞, 0) at infinity. Then we set µ = (1 − κ)−1
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Akira Kaneko
and call it the multiplicity of ν. A more exact definition may be given via the Newton polygon, transforming the point at infinity to the origin: Set σ = 1/s, τ = t/s and let
q(σ, τ ) = s−mQ(s, t) s=1/σ,t=τ/σ.
Factorize it as a polynomial of τ with coefficients in Oσ,0 at 0. Then µ is the inverse of the minimum value with respect to the irreducible factors of the leading powers of the Puiseux expansions of the roots of them.
Notice that the irregularity employed here is the mildest one, in comparison with the strongest one which is usually used e.g. to define the hyperbolicity.
Our main result here is the following
Theorem 1.4. Let K be a compact set contained in a hyperplane ν · x = 0. Assume further that every irreducible component of P (ζ) has ν as characteristic direction of irregularity ≤ µ. Then
EP(s)(Ω \ K)/EP(s)(Ω) = 0, if s ≤ µ/(µ − 1),
EP{s}(Ω \ K)/EP{s}(Ω) = 0, if s < µ/(µ − 1).
Corollary 1.5. Assume that every irreducible component of P (ζ) is non-elliptic. Then the isolated singularities of solutions of class E1+ are
removable, that is,
(1.14)
EP1+(Ω \ {0})/EP1+(Ω) = 0.
Remark that this sufficient condition on P (D) is the same as the one for the removablilty of isolated singularities of real analytic solutions given in [Kn1]–[Kn2].
Although the proof of Theorem 1.4 is almost a literal translation of Grushin’s original article [G2] for the removability of isolated singularities of C∞ solutions, we shall reproduce it here in detail, because it is nevertheless