Formal Gevrey Theory for Singular First Order Quasi-Linear Partial Differential Equations

Formal Gevrey Theory for Singular First Order Quasi-Linear Partial Differential Equations

By

MasakiHibino∗

Abstract

This paper is concerned with the existence, the uniqueness, convergence and divergence of formal power series solutions of singular first order quasi-linear partial differential equations. Firstly we give the condition under which the formal solution exists uniquely. However, this formal power series solution does not necessarily con- verge. So we characterize the rate of divergence of the formal solution via the Gevrey order of formal power series. The Gevrey orders of formal solutions are determined by the Newton polyhedrons of nonlinear partial differential operators.

§1. Introduction and Main Result

In this paper we are concerned with formal power series solutions of the following first order quasi-linear partial differential equation:

(1.1)

P u(x)≡^d

i=1

a_i(x, u(x))D_iu(x) =f(x, u(x)), u(0) = 0,

x= (x₁, . . . , x_d)∈C^d, u∈C, D_i =∂_xi = ∂

∂x_i,

where coefficientsa_i(x, u) (i= 1, . . . , d) andf(x, u) are holomorphic in a neigh- borhood of (x, u) = (0,0).

Communicated by T. Kawai. Recieved October 1, 2004. Revised May 30, 2005, Septem- ber 5, 2005.

2000 Mathematics Subject Classification(s): Primary 35C10; Secondary 35A10.

∗Research Fellow of the Japan Society for the Promotion of Science

Department of Mathematics, Meijo University, Tempaku, Nagoya, Aichi 468-8502, Japan.

Current address: Department of Intelligent Mechanical Engineering, Okayama University of Science, 1-1 Ridai-cho, Okayama, 700-0005, Japan

e-mail: hibino@are.ous.ac.jp

We shall study the case where

(1.2) a_i(0,0) = 0 for all i= 1, . . . , d,

which is called a singular or degenerate case. Moreover as a compatibility condition we assume the following:

(1.3) f(0,0) = 0.

In the following we always assume (1.2) and (1.3). We remark that by (1.2) Cauchy-Kowalevsky’s theorem is not available.

We have two purposes. The first one is to prove the existence and the uniqueness of the formal power series solution u(x) =

|α|≥1u_αx^α (α = (α₁, . . . , α_d)∈N^d, N={0,1,2, . . .}, |α|=α₁+· · ·+α_d, x^α =x₁^α1· · ·x_d^αd) centered at the origin for the singular equation (1.1). As we will see later, we can prove it under some condition on the Jacobi matrix at the origin of the vector fieldC^dx→(a₁(x,0), . . . , a_d(x,0))∈C^d. However, this formal power series solution u(x) does not necessarily converge. So we would like to obtain the rate of divergence, which is called the Gevrey order, of the formal solution (cf. Definition 1.1). This is the second purpose of this paper. Hibino [2][3]

studied the same problems for linear equations and semi-linear equations. In this paper we generalize these studies to quasi-linear equations.

The content of this paper is as follows: In §1.1 we introduce the result in [3] for semi-linear equations. To state the result (Theorem 1.1), we give the definition of the Gevrey order and explain the Poincar´e condition, which assures the existence and the uniqueness of the formal solution. In§1.2, using the notation provided in§1.1, we give the main result in this paper (Theorem 1.2). Under one additional condition (cf. (1.6)), we will obtain the same result as that of [3]. In§1.3 we introduce literature studying related topics. In§2 we divide equations into seven classes, and for each class we give a precise Gevrey order of the formal solution, which is determined by the Newton Polyhedron of a quasi-linear differential operator (Theorem 2.1). Theorem 1.2 is obtained as a corollaly of Theorem 2.1. The proof of Theorem 2.1 is done through §4,

§5 and§6 by using the contraction mapping principle in Banach spaces which consist of formal power series. The Banach spaces employed in the proof will be introduced in§3.

§1.1. Result for Semi-Linear Equations

In this subsection we state the theorem obtained in [3] for semi-linear equations. First we give the definition of the Gevrey order, which gives the rate of divergence of formal power series.

Definition 1.1. Letu(x) =

α∈N^du_αx^αbe a formal power series cen- tered at the origin. We say thatu(x)is of Gevrey-{s} class(s= (s₁, . . . , s_d)∈ R^d) if the power series

Bs[u](ξ)≡

α∈N^d

u_α ξ^α (α!)^s−1(d)

converges in a neighborhood ofξ= 0, where 1^(d)= ( d

1, . . . ,1),s−1^(d)= (s₁− 1, . . . , s_d−1) and (α!)^s−1(d) = (α₁!)^s1−1· · ·(α_d!)^sd−1. G^{s} denotes the set of all formal power series being of Gevrey-{s}class. In particular,u(x)∈G^{1(d)} if and only if u(x) is a convergent power series nearx= 0.

Now let us consider the following singular first order semi-linear partial differential equation:

(1.4) P u(x) ≡ d i=1

a_i(x)D_iu(x) =f(x, u(x)), u(0) = 0,

where coefficients a_i(x) (i= 1, . . . , d) and f(x, u) are holomorphic in a neigh- borhood ofx= 0 and (x, u) = (0,0), respectively. Moreover it is assumed that a_i(0) = 0 for alli= 1, . . . , dand (1.3).

LetD_xa(0) = (D_ja_i(0))i,j=1,...,d be the Jacobi matrix at the origin of the vector field x→(a₁(x), . . . , a_d(x)) and let its Jordan canonical form be

(1.5)









 A

B₁ . ..

B_k O_p











where

A=







 λ₁ δ₁

λ₂ . ..

. .. δ_m−1 λ_m









, B_h=







 0 1

0 . .. . .. 1

0









nh

,

λ_i= 0 (i= 1, . . . , m), δ_i= 0 or 1

(i= 1, . . . , m−1), h= 1, . . . , k,

andO_pis a zero-matrix of orderp(m,k,p≥0;n_h≥2;m+n₁+· · ·+n_k+p=d).

Let us assume the following condition (Po) according to the value of m (“Po” derives from Poincar´e):

(Po)







m i=1

λ_iα_i−f_u(0,0)

> c|α| for all α∈N^m (if m≥1),

f_u(0,0)= 0 (if m= 0),

where f_u(0,0) = (∂f /∂u)(0,0), and c is a positive constant independent of α∈N^m.

The main result in [3] is stated as follows:

Theorem 1.1 ([3]). Under the condition (Po), the equation(1.4)has a unique formal power series solution u(x) =

|α|≥1u_αx^α. Furthermore u(x)∈ G{2N,...,2N}, where

N =









max{n₁, . . . , n_k} (if k≥1),

1 (if k= 0 andp≥1),

1

2 (if k=p= 0).

Therefore in the case k=p= 0 the formal solution converges. In other cases it diverges in general.

As was mentioned before, the purpose of this paper is to generalize the above result to quasi-linear equations.

Remark 1.1. The estimates of the Gevrey order given in Theorem 1.1 could be improved if the coefficients a_i(x) have a higher order root for x = 0. For example, in the case m = k = 0 and p ≥ 1, the formal solution u(x) =_∞

n=1(2n−3)!!x²ⁿ⁻¹ ((−1)!! = 1) of the ordinary differential equation

−x³(d/dx)u(x) +u(x) =xbelongs toG^{3/2}. Such a precise Gevrey order will be given in Theorem 2.1.

§1.2. Main Result Let us state the main result in this paper.

LetJbe the Jacobi matrix at the origin of the vector fieldx→(a₁(x,0), . . . , a_d(x,0)), that is,J = (D_ja_i(0,0))i,j=1,...,d, and let us write its Jordan canonical form as (1.5).

Similarly to Theorem 1.1, we assume the condition (Po). Moreover we assume the following additional condition:

(1.6) ∂a_i

∂u(0,0) = 0 for all i= 1, . . . , d.

Now the main result in this paper is stated as follows:

Theorem 1.2. Under the conditions(Po)and(1.6), the equation (1.1) has a unique formal power series solution u(x) =

|α|≥1u_αx^α. Furthermore u(x)∈G{2N,...,2N}, whereN is same as in Theorem1.1.

In order to prove Theorem 1.2, as a first step we shall transform the equa- tion (1.1) in the next section. For that transformed equation we can obtain the precise Gevrey order in individual variables of the formal solution (Theorem 2.1). We prove the unique existence of the formal solution and its Gevrey order separately. However, the proof of the unique existence is quite similar to that for semi-linear equations (cf. §6 in [3]), so in this paper we omit it. Admitting the unique existence of the formal solution, we will prove its Gevrey order in

§4 (the casem= 0 andk=p= 0),§5 (the casem= 0) and§6 (the casem= 0 and (k, p)= (0,0)) by adopting the contraction mapping principle in Banach spaces introduced in§3. This route of the proof is same as that in [3], but the Banach spaces employed in the proof is slightly different from those used in [3]

(cf. Definition 3.1 and Remark 3.1).

§1.3. Some Remarks on Related Topics

The studies in this paper and [2][3] are inspired by the study in ¯Oshima [9]. He studied a characterization of the kernel and the cokernel of the linear mapping

P₀:O → O

P₀u(x) = d i=1

a_i(x)D_iu(x) +b(x)u(x)

,

whereOis the set of holomorphic functions at the origin, anda_i(x),b(x)∈ O. He studied the case m ≥ 1 and k = 0 in our notation, and obtained the condition under which the formal solution converges. That condition is called the simple ideal condition (cf. Remark 2.2). However, as mentioned in our theorem, whenm≥1,k= 0 andp≥1, if we remove the simple ideal condition, the formal solution diverges in general and it belongs toG^{2,...,2}.

The problem of convergence and divergence for formal solutions has been studied by many mathematicians. Higher order equations are studied by Miyake [5], Miyake-Hashimoto [6] and Yamazawa [13]. Nonlinear equations are studied in G´erard-Tahara [1], Miyake-Shirai [7] and ¯Ouchi [10]. In ¯Ouchi [10], the ex- istence of asymptotic solutions is also dealt with. The study of Yamazawa [12]

for semi-linear equations gives a different viewpoint from the author’s study [3]. Moreover for linear equations, Kashiwara-Kawai-Sj¨ostrand [4] and Miyake- Yoshino [8] give different characterizations of convergence of formal solutions.

We can consider these studies to be the generalizations of ¯Oshima’s study.

However, in these studies the case k≥1 has not been studied. In this sense, [2][3] and Theorem 1.2 give a different type of generalization of ¯Oshima [9].

Recently, Sibuya [11] studied greatly general higher order nonlinear ordi- nary differential equations with one parameter:

F

x, u,du

dx, . . . ,d^lu dx^l, ε

= 0,

where x, ε∈C. [11] assumed the existence of formal solutions which take the forms of

(1.7) u(x, ε) =

u_n(x)εⁿ,

whereu_n(x) are holomorphic in a common domain, and gave the Gevrey orders ofuwith respect to the parameterε. Our equation (1.1) becomes a first order ordinary differential equation depending upon several parameters if all but one coefficient is identically zero. However, in this case our result (in case of one parameter) does not follow from that of [11]. In general, formal solutions can not be written in the forms of (1.7).

§2. Reduction of Equation and Newton Polyhedron

In order to prove Theorem 1.2 we shall transform the equation (1.1) by a linear transform of independent variables which reduces the Jacobi matrix J to its Jordan canonical form. By (1.2), (1.3) and (1.6), a reduced equation is written as follows according to the values ofm,kand p:

Case (i) m≥1,k≥1,p≥1:

Λu(x, y, z) = (P+P+P+P)u(x, y, z) (2.1)

+g₀(x, y, z) +g(x, y, z, u(x, y, z)), u(0,0,0) = 0,

x= (x₁, . . . , x_m)∈C^m, y= (y¹, . . . , y^k)∈C^n1+···+nk, y^h= (y^h₁, . . . , y^h_n

h)∈C^nh(h= 1, . . . , k), z= (z₁, . . . , z_p)∈C^p,

where

(2.2) Λ=

m i=1

λ_ix_i ∂

∂x_i −f_u(0,0),

and

Pu=−^m−1

i=1

δ_ix_i+1∂u

∂x_i + m i=1

finite

|α|+|β|+|γ|+r≥2

|α|≥1

c_iαβγr(x, y, z, u)x^αy^βz^γu^r ∂u

∂x_i,

Pu= k h=1

nh

jh=1

finite

|α|+|β|+|γ|+r≥2

|α|≥1

d^h_j

hαβγr(x, y, z, u)x^αy^βz^γu^r ∂u

∂y^h_j

h

+ p q=1

finite

|α|+|β|+|γ|+r≥2

|α|≥1

e_qαβγr(x, y, z, u)x^αy^βz^γu^r ∂u

∂z_q,

Pu=− k h=1

nh−1 jh=1

y^h_j

h+1

∂u

∂y^h_j

h

+ k h=1

nh

jh=1

finite

|β|+|γ|+r≥2

d^h_j

hβγr(x, y, z, u)y^βz^γu^r ∂u

∂y^h_j

h

+ p q=1

finite

|β|+|γ|+r≥2

e_qβγr(x, y, z, u)y^βz^γu^r ∂u

∂z_q,

Pu= m i=1

finite

|β|+|γ|+r≥2

c_iβγr(x, y, z, u)y^βz^γu^r ∂u

∂x_i

(x^α =x₁^α1· · ·x_m^αm, y^β = (y¹)^β1· · ·(y^k)^βk, (y^h)^βh = (y^h₁)^βh1· · ·(y^h_n

h)^βhnh (h= 1, . . . , k),z^γ =z₁^γ1· · ·z_p^γp). g₀andgare holomorphic at the origin which satisfyg₀(0,0,0) = 0 andg(x, y, z,0)≡0,g_u(0,0,0,0) = 0, respectively. In the above expressions, all coefficientsc_iαβγr(x, y, z, u), etc., are holomorphic at the origin, and none of them vanish at the origin unless they vanish identically. In the following expressions, we assume the same conditions for those functions appearing in the coefficients.

Case (ii) m≥1,k≥1,p= 0:

(2.3) Λu(x, y) = (P+P+P+P)u(x, y) +g₀(x, y) +g(x, y, u(x, y)), u(0,0) = 0,

where g₀ and g are holomorphic at the origin which satisfy g₀(0,0) = 0 and g(x, y,0)≡0,g_u(0,0,0) = 0, respectively. The linear partial differential oper- atorΛis same as (2.2), and

Pu=−^m−1

i=1

δ_ix_i+1∂u

∂x_i + m i=1

finite

|α|+|β|+r≥2

|α|≥1

c_iαβr(x, y, u)x^αy^βu^r ∂u

∂x_i,

Pu= k h=1

nh

jh=1

finite

|α|+|β|+r≥2

|α|≥1

d^h_j

hαβr(x, y, u)x^αy^βu^r ∂u

∂y^h_j

h

,

Pu=− k h=1

nh−1 jh=1

y^h_j

h+1

∂u

∂y^h_j

h

+ k h=1

nh

jh=1

finite

|β|+r≥2

d^h_j

hβr(x, y, u)y^βu^r ∂u

∂y^h_j

h

,

Pu= m i=1

finite

|β|+r≥2

c_iβr(x, y, u)y^βu^r ∂u

∂x_i. Case (iii) m≥1,k= 0, p≥1:

(2.4) Λu(x, z) = (P+P+P+P)u(x, z) +g₀(x, z) +g(x, z, u(x, z)), u(0,0) = 0,

whereg₀andgare holomorphic at the origin withg₀(0,0) = 0 andg(x, z,0)≡0, g_u(0,0,0) = 0, respectively. The linear partial differential operator Λis same as (2.2), and

Pu=−

m−1

i=1

δ_ix_i+1∂u

∂x_i + m i=1

finite

|α|+|γ|+r≥2

|α|≥1

c_iαγr(x, z, u)x^αz^γu^r ∂u

∂x_i,

Pu= p q=1

finite

|α|+|γ|+r≥2

|α|≥1

e_qαγr(x, z, u)x^αz^γu^r ∂u

∂z_q,

Pu= p q=1

finite

|γ|+r≥2

e_qγr(x, z, u)z^γu^r ∂u

∂z_q,

Pu= m i=1

finite

|γ|+r≥2

c_iγr(x, z, u)z^γu^r ∂u

∂x_i. Case (iv) m≥1,k=p= 0:

Λu(x) =−^m−1

i=1

δ_ix_i+1∂u

∂x_i(x) + m i=1

finite

|α|+r≥2

c_iαr(x, u(x))x^αu(x)^r ∂u

∂x_i(x) +g₀(x) +g(x, u(x)),

u(0) = 0, (2.5)

where g₀ and g are holomorphic at the origin with g₀(0) = 0 andg(x,0)≡0, g_u(0,0) = 0, respectively. Λ is same as (2.2).

Case (v) m= 0,k≥1,p≥1:

(2.6) f_u(0,0)·u(y, z) =Pu(y, z) +g₀(y, z) +g(y, z, u(y, z)), u(0,0) = 0, where g₀ and g are holomorphic at the origin which satisfy g₀(0,0) = 0 and g(y, z,0)≡0,g_u(0,0,0) = 0, respectively, and

Pu= k h=1

nh−1 jh=1

y^h_j

h+1

∂u

∂y^h_j

h

+ k h=1

nh

jh=1

finite

|β|+|γ|+r≥2

d^h_j

hβγr(y, z, u)y^βz^γu^r ∂u

∂y^h_j

h

+ p q=1

finite

|β|+|γ|+r≥2

e_qβγr(y, z, u)y^βz^γu^r ∂u

∂z_q. Case (vi) m= 0,k≥1,p= 0:

(2.7) f_u(0,0)·u(y) =Pu(y) +g₀(y) +g(y, u(y)), u(0) = 0,

where g₀ and g are holomorphic at the origin which satisfy g₀(0) = 0 and g(y,0)≡0,g_u(0,0) = 0, respectively, and

Pu= k h=1

nh−1 jh=1

y^h_j

h+1

∂u

∂y^h_j

h

+ k h=1

nh

jh=1

finite

|β|+r≥2

d^h_j

hβr(y, u)y^βu^r ∂u

∂y^h_j

h

.

Case (vii) m=k= 0,p≥1:

(2.8) f_u(0,0)·u(z) =Pu(z) +g₀(z) +g(z, u(z)), u(0) = 0,

where g₀ and g are holomorphic at the origin withg₀(0) = 0 andg(z,0)≡0, g_u(0,0) = 0, respectively, and

Pu= p q=1

finite

|γ|+r≥2

e_qγr(z, u)z^γu^r ∂u

∂z_q.

Now we shall study the equations (2.1), (2.3), (2.4), (2.5), (2.6), (2.7) and (2.8).

In order to give the Gevrey orders in an individual variable for formal solutions of the above equations, we study the Newton polyhedrons of nonlinear partial differential operators.

Newton Polyhedron. Let Lu(ξ) =

finite

α,β∈N^d,|β|≥1

a_αβr(ξ, u(ξ))ξ^αu(ξ)^rD_ξ^βu(ξ)

(ξ= (ξ₁, . . . , ξ_d), D_ξ^β = (∂/∂ξ₁)^β1· · ·(∂/∂ξ_d)^βd) be a nonlinear partial differ- ential operator which is linear with respect to derivatives, where all coefficients a_αβr(ξ, u) are holomorphic at the origin and do not vanish at the origin unless they vanish identically.

We defineQ(α, β) (⊂R^d+1) by

Q(α, β) ={(X,W) = (X₁, . . . ,Xd,W)∈R^d+1;

Xi≥α_i−β_i (i= 1, . . . , d), W ≤ |β|}

and define Q(ξ^αu(ξ)^rD_ξ^βu(ξ)) by Q(ξ^αu(ξ)^rD_ξ^βu(ξ)) =

d i=1

Q(α+re_i, β),

where e_i = (δ_i1, δ_i2, . . . , δ_id) (δ_ii: Kronecker’s delta). We remark that Q(ξ^αD_ξ^βu(ξ)) =Q(α, β). Let us define the Newton polyhedron N(L) of the operator Lby

N(L) =









 Ch





(α,β,r) withaαβr≡0

Q(ξ^αu(ξ)^rD_ξ^βu(ξ))



(ifL= 0),

Q(0,0) (ifL= 0),

where ChAdenotes the convex hull of a setA⊂R^d+1.

Now we shall apply the above general definition to our differential operators P+P+P+P(Cases (i), (ii) and (iii)) andP (Cases (v), (vi) and (vii)).

We remark that the correspondence of variables between (x, y, z) andξis given by

ξ Case (i) (x, y, z) Case (ii) (x, y) Case (iii) (x, z) Case (iv) — Case (v) (y, z) Case (vi) y Case (vii) z

In order to state the main theorem in this section, we shall define the setsS_i (i= 1, 2, 3, 5, 6, 7), S_j,S_j,S_j, S_j, S_j (j= 1, 2, 3) whose elements give the Gevrey orders of formal solutions.

Case (i) We define Π₁(ρ, σ, τ) and Π₁(ρ, σ, τ) ((ρ, σ, τ) ∈ [1,+∞)^d, ρ = (ρ₁, . . . , ρ_m) ∈ [1,+∞)^m, σ = (σ¹, . . . , σ^k) ∈ [1,+∞)^n1+···+nk, σ^h = (σ^h₁, . . . , σ^h_nh)∈[1,+∞)^nh,τ= (τ₁, . . . , τ_p)∈[1,+∞)^p) by

Π₁(ρ, σ, τ) =

(X,Y,Z,W)∈R^d+1;

(ρ−1^(m))· X + (σ−1^{(n1+···+nk)})· Y+ (τ−1^(p))· Z − W ≥ −1

(X = (X₁, . . . ,Xm), Y = (Y¹, . . . ,Y^k), Y^h = (Y^h₁, . . . ,Y^hnh) (h= 1, . . ., k), Z = (Z₁, . . . ,Z_p);A·B means the scalar product ofAandB) and

Π₁(ρ, σ, τ) =

(X,Y,Z,W)∈R^d+1;

(ρ−1^(m))· X+ (σ−1^{(n1+···+nk)})· Y+ (τ−1^(p))· Z − W ≥0

, respectively, and defineS₁,S₁,S₁,S₁,S₁, andS₁ as follows:

S₁={(ρ, σ, τ)∈[1,+∞)^d; N(P)⊂Π₁(ρ, σ, τ)}, S₁ ={(ρ, σ, τ)∈[1,+∞)^d; N(P)⊂Π₁(ρ, σ, τ)}, S₁={(ρ, σ, τ)∈[1,+∞)^d; N(P)⊂Π₁(ρ, σ, τ)},

S₁={(ρ, σ, τ)∈[1,+∞)^d; N(P)⊂Π₁(ρ, σ, τ)}, S₁ ={(ρ, σ, τ)∈[1,+∞)^d; N(P)⊂Π₁(ρ, σ, τ)}, S₁={(ρ, σ, τ)∈[1,+∞)^d; N(P)⊂Π₁(ρ, σ, τ)}. Case (ii) We defineΠ₂(ρ, σ) andΠ₂(ρ, σ) ((ρ, σ)∈[1,+∞)^d) by Π₂(ρ, σ) =

(X,Y,W)∈R^d+1; (ρ−1^(m))·X+(σ−1^{(n1+···+nk)})·Y −W ≥ −1

and Π₂(ρ, σ) =

(X,Y,W)∈R^d+1; (ρ−1^(m))·X+ (σ−1^{(n1+···+nk)})·Y −W ≥0

, respectively, and defineS₂,S₂,S₂,S₂,S₂ andS₂as follows:

S₂={(ρ, σ)∈[1,+∞)^d; N(P)⊂Π₂(ρ, σ)}, S₂ ={(ρ, σ)∈[1,+∞)^d; N(P)⊂Π₂(ρ, σ)}, S₂={(ρ, σ)∈[1,+∞)^d; N(P)⊂Π₂(ρ, σ)},

S₂={(ρ, σ)∈[1,+∞)^d; N(P)⊂Π₂(ρ, σ)}, S₂ ={(ρ, σ)∈[1,+∞)^d; N(P)⊂Π₂(ρ, σ)}, S₂={(ρ, σ)∈[1,+∞)^d; N(P)⊂Π₂(ρ, σ)}.

Case (iii) We defineΠ₃(ρ, τ) andΠ₃(ρ, τ) ((ρ, τ)∈[1,+∞)^d) by Π₃(ρ, τ) =

(X,Z,W)∈R^d+1; (ρ−1^(m))· X + (τ−1^(p))· Z − W ≥ −1

and

Π₃(ρ, τ) =

(X,Z,W)∈R^d+1; (ρ−1^(m))· X + (τ−1^(p))· Z − W ≥0

, respectively, and defineS₃,S₃,S₃,S₃,S₃,S₃ as follows:

S₃={(ρ, τ)∈[1,+∞)^d; N(P)⊂Π₃(ρ, τ)}, S₃ ={(ρ, τ)∈[1,+∞)^d; N(P)⊂Π₃(ρ, τ)}, S₃={(ρ, τ)∈[1,+∞)^d; N(P)⊂Π₃(ρ, τ)},

S₃={(ρ, τ)∈[1,+∞)^d; N(P)⊂Π₃(ρ, τ)}, S₃ ={(ρ, τ)∈[1,+∞)^d; N(P)⊂Π₃(ρ, τ)}, S₃={(ρ, τ)∈[1,+∞)^d; N(P)⊂Π₃(ρ, τ)}. Case (v) We defineΠ₅(σ, τ) ((σ, τ)∈[1,+∞)^d) by Π₅(σ, τ) =

(Y,Z,W)∈R^d+1; (σ−1^{(n1+···+nk)})· Y+ (τ−1^(p))· Z − W ≥0

, and define S₅by

S₅={(σ, τ)∈[1,+∞)^d; N(P)⊂Π₅(σ, τ)}. Case (vi) We defineΠ₆(σ) (σ∈[1,+∞)^d) by

Π₆(σ) =

(Y,W)∈R^d+1; (σ−1^{(n1+···+nk)})· Y − W ≥0

, and define S₆by

S₆={σ∈[1,+∞)^d; N(P)⊂Π₆(σ, τ)}. Case (vii) We defineΠ₇(τ) (τ ∈[1,+∞)^d) by

Π₇(τ) =

(Z,W)∈R^d+1; (τ−1^(p))· Z − W ≥0

, and define S₇by

S₇={τ ∈[1,+∞)^d; N(P)⊂Π₇(τ)}. Then we obtain the following theorem.

Theorem 2.1. In Case (i) (resp. (ii), (iii), (iv), (v), (vi) and (vii)), under the condition (Po) the equation (2.1) (resp. (2.3), (2.4), (2.5), (2.6), (2.7) and (2.8)) has a unique formal power series solution. Furthermore the formal solution belongs to G^{s} if ssatisfies the following condition:

Case (i) P= 0 ⇒s= (ρ, σ, τ)∈S₁∩S₁∩S₁, P= 0 ⇒s= (ρ, σ, τ)∈S₁∩S₁∩S₁,

P,P= 0 ⇒s= (ρ, σ, τ)∈S₁∩S₁∩ {(S₁∩S₁)∪(S₁ ∩S₁)}; Case (ii) P= 0 ⇒s= (ρ, σ)∈S₂∩S₂∩S₂,

P= 0 ⇒s= (ρ, σ)∈S₂∩S₂∩S₂,

P,P= 0 ⇒s= (ρ, σ)∈S₂∩S₂∩ {(S₂ ∩S₂)∪(S₂ ∩S₂)}; Case (iii) P= 0 ⇒s= (ρ, τ)∈S₃∩S₃∩S₃,

P= 0 ⇒s= (ρ, τ)∈S₃∩S₃∩S₃,

P,P= 0 ⇒s= (ρ, τ)∈S₃∩S₃∩ {(S₃∩S₃)∪(S₃ ∩S₃)}; Case (iv) s= 1^(d);

Case (v) s= (σ, τ)∈S₅; Case (vi) s=σ∈S₆; Case (vii) s=τ∈S₇.

On the concrete method of determining Gevrey orders, for example, see [2] (see also Lemmas 5.1, 6.1 and Remarks 5.2, 6.2).

Remark 2.1. We can easily see that the followings₀always satisfies the condition in Theorem 2.1 for each case:

Case (i) s₀= (ρ₀, σ₀, τ₀), Case (ii) s₀= (ρ₀, σ₀), Case (iii) s₀= (ρ₀, τ₀), Case (iv) s₀= 1^(d), Case (v) s₀= (σ₀, τ₀), Case (vi) s₀=σ₀, Case (vii) s₀=τ₀, where ρ₀ = (

m

N+ 1, . . . , N+ 1), σ₀ = (σ¹₀, . . . , σ^k₀), σ^h₀ = (N + 1, N + 2, . . . , N+n_h) (h= 1,. . .,k), τ₀ = (

p

N+ 1, . . . , N+ 1). Therefore by a linear transform of independent variables again we obtain Theorem 1.2 from Theorem 2.1 and the next Lemma 2.1. Thus the proof of Theorem 1.2 is reduced to that of Theorem 2.1.

Lemma 2.1 ([2]). Let u(x) =

α∈N^du_αx^α∈G{s,s,...,s} (s≥1). Then for any linear transformL:C^d→C^d, it holds thatv(y) :=u(Ly)∈G{s,s,...,s}.

From Theorem 2.1, on the existence of holomorphic solutions we obtain the following corollary.

Corollary 2.1. Let us consider Case (iii), and let us assume the fol- lowing condition:

(I) P=P= 0 or

(II) P=P= 0.

Then under the condition (Po) the equation (2.4) has a unique holomorphic solution at the origin.

Let us consider Case(iv)and let us assume the condition (Po). Then the equation (2.5) has a unique holomorphic solution at the origin.

Remark 2.2. In Corollay 2.1, the condition (II) corresponds to the simple ideal condition in ¯Oshima [9].

From §4 we shall start the proof of Theorem 2.1. In §4 we shall deal with Case (iv), and in this case the proof is very simple. In §5 we shall deal with Cases (v), (vi) and (vii). However, these three cases are proved by an essentially same method, so the proof will be done only in Case (v). Finally, in §6 we prove the theorem for Cases (i), (ii) and (iii). In these cases also, the theorem is proved by an essentially same method, so we will deal with only Case (i). We remark that methods of proof employed in§5 and§6 are different.

§3. Banach Spaces G^{s}_0,k(R) andG^{s_0,k^1,s2}(R¹, R²)

Theorem 2.1 is proved by a contraction mapping principle in Banach spaces which consist of formal power series. For this purpose in this section we shall define two types of Banach spaces necessary in the proof, and we shall prove some fundamental facts needed later.

Definition 3.1. (1) Let s= (s₁, . . . , s_d)∈R+d (R+ ={r ∈R; r ≥ 0}) and R = (R₁, . . . , R_d) ∈ (R+\ {0})^d, and let k be a real number which satisfies k ≤ min{s₁, . . . , s_d}. The space of formal power series G^{s}_0,k(R) is defined as follows:

We say thatu(x) =

α∈N^du_αx^α belongs toG^{s}_0,k(R) ifu(0) = 0 and u^{s}_k,R≡

|α|≥1

|u_α| |α|!

(s·α−k)!R^α<+∞,

where l! = Γ(l+ 1) for l ≥ 0. We remark that k ≤ min{s₁, . . . , s_d} implies s·α−k≥0 for allαsuch that|α| ≥1.

(2) Let (s¹, s²) = (s¹₁, . . . , s¹_d1, s²₁, . . . , s²_d2)∈R+d1+d2 and (R¹, R²)∈ (R+\ {0})^d1+d2 (R¹= (R¹₁, . . . , R¹_d1),R² = (R²₁, . . . , R²_d2)), and letk be a real number which satisfiesk≤min{s¹₁, . . . , s¹_d1, s²₁, . . . , s²_d2}. The space of formal power seriesG^{s_0,k^1,s2}(R¹, R²) is defined as follows:

We say thatu(x, y) =

(α,β)∈N^d1+d2u_αβx^αy^β belongs toG^{s_0,k^1,s2}(R¹, R²) ifu(0,0) = 0 and

|||u|||^{s_k,R^1,s1,R^2}2 ≡

|α|+|β|≥1

|u_αβ| |α|!|β|!

(s¹·α+s²·β−k)!(R¹)^α(R²)^β<+∞. ThenG^{s}_0,k(R) andG^{s_0,k^1,s2}(R¹, R²) are Banach spaces equipped with the norms · ^{s}_k,R and||| · |||^{s_k,R^1,s1,R^2}2, respectively.

Remark 3.1. In [3], we introduced the spacesG^{s}_0,0(R) andG^{s_0,0^1,s2}(R¹, R²). In the proof of the theorem for quasi-linear equations (in particular, when we prove divergence of the formal solution), the spaces G^{s}_0,k(R) and G^{s_0,k^1,s2}(R¹, R²) for a positivekplay significant roles (cf. §5 and §6).

Lemma 3.1. (1) Let s∈R+d and let k≤min{s₁, . . . , s_d}. Then for all R∈(R+\ {0})^d it holds that

G^{s}_0,k(R)⊂G^{s}.

(2) Let (s¹, s²)∈ R+d1+d2 and let k ≤min{s¹₁, . . . , s¹_d1, s²₁, . . . , s²_d2}. Then for all (R¹, R²)∈(R+\ {0})^d1+d2 it holds that

G^{s_0,k^1,s2}(R¹, R²)⊂G^{s1,s2}.

Proof. (1): In general it follows from Cauchy’s integral formula that a formal power series u(x) =

α∈N^du_αx^α belongs to G^{s} if and only if there exist some positive constants AandB such that

|u_α| ≤AB^|α|(α!)^s−1(d) for all α∈N^d. Now letu(x) =

|α|≥1u_αx^α∈G^{s}_0,k(R). Then we have

|u_α| ≤ u^{s}_k,R(s·α−k)!

|α|! · 1

R^α ≤ u^{s}_k,R(s·α−k)!

(α!)^s · 1

R^α ·(α!)^s−1(d). By Stirling’s formula there exist some positive constantsA andB such that

(s·α−k)!

(α!)^s ≤A·B^|α| for all α(|α| ≥1).