ON THE CAUCHY PROBLEM IN A CLASS OF ENTIRE FUNCTIONS

(1)

SINGULARITIES AND DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 33

INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 1996

ON THE CAUCHY PROBLEM IN A CLASS OF ENTIRE FUNCTIONS

IN SEVERAL VARIABLES

E U G E N I L E I N A R T A S Krasnoyarsk State University Svobodny 79, 660062 Krasnoyarsk, Russia

Abstract. We study the integral representation of solutions to the Cauchy problem for a differential equation with constant coefficients. The Cauchy data and the right-hand of the equation are given by entire functions on a complex hyperplane of C

ⁿ⁺¹

. The Borel transformation of power series and residue theory are used as the main methods of investigation.

1. Introduction. For holomorphic partial differential equations the local theory of Cauchy problem is well developed. In the non-characteristic case the classical Cauchy- Kovalevskaya theorem states existence and uniqueness of analytic solutions. Globally, if we have the entire Cauchy data on a hyperplane, the Cauchy-Kovalevskaya theorem can in certain cases be extended to yield entire solutions (cf. results by M.Miyake [1] and J.Persson [2]).

Recently, Sternin and Shatalov have given explicit solutions of the global Cauchy problem in the constant coefficient case with Cauchy data on an arbitrary analytic hy- persurface in terms of the Radon-Laplace integral transform (see [3]).

In this paper a new integral representation for the solutions to a class of Cauchy problems is obtained. We assume that Cauchy data and the right-hand of a partial differential equation are entire functions. As an example we consider the case when the characteristic polynomial P (τ, ξ) is homogeneous polynomial in two variables (τ, ξ) ∈ C

²

.

2. Notation and definitions. We will work in C

ⁿ⁺¹

using the following variables:

x = (x

₁

, . . . , x

_n

) ∈ C

ⁿ

, ξ = (ξ

₁

, . . . , ξ

_n

) ∈ C

ⁿ

, t ∈ C, τ ∈ C.

C

ⁿx

is considered as a linear subspace of C

ⁿ⁺¹

, i.e. C

ⁿx

= {(t, x) ∈ C

ⁿ⁺¹

: t = 0}.

α = (α

1

, . . . , α

n

) ∈ N

ⁿ

and β = (β

1

, . . . , β

n

) are multiindices, |α| = α

1

+ . . . + α

n

,

1991 Mathematics Subject Classification: 32A25, 32A27.

The paper is in final form and no version of it will be published elsewhere.

[189]

(2)

190

E. LEINARTAS

x

^α

= x

^α₁¹

. . . x

^α_nⁿ

. For differential operators we use the following notation:

D

t

= ∂

∂t , D

_x^α

= ∂

^|α|

u

∂x

^α₁¹

. . . ∂x

^αnⁿ

.

We treat mth order, linear PDEs with constant coefficients of the following type:

(1) D

^m_t

u +

m−1

X

k=0

X

|α|≤k

a

k,α

D

^m−k−1_t

D

_x^α

u = Φ(t, x) or for short:

(∗) P (D)u = Φ.

The corresponding symbol for P is P (τ, ξ) = τ

^m

+

m−1

X

k=0

X

|α|≤k

a

_k,α

ξ

^α

τ

^m−k−1

=

m

X

µ=0

b

_µ

( ξ ) τ

^m−µ

,

where τ ∈ C, ξ ∈ C

ⁿ

, b

0

(ξ) ≡ 1, b

µ

(ξ) = P

|α|≤µ

a

m−µ,α

ξ

^α

, µ = 1, . . . , m. The equation P (τ, ξ) = 0 is called characteristic with respect to (1).

We can now formulate the following Cauchy problem. Suppose that Φ(t, x) is an entire function in C

ⁿ⁺¹

and that we have entire functions v

k

(x) in C

ⁿx

. We seek the unique (entire) solution u(t, x) satisfying

(∗) P (D)u = Φ and

(∗∗) D

_t^k

u = v

_k

(x), k = 1, . . . , m − 1, for t = 0.

Note that since the equation (1) is normal (i.e. b

0

(ξ) = 1), the Cauchy-Kovalevskaya theorem yields a unique, local solution u(t, x) of (∗), (∗∗). If we find an entire solution

˜

u(t, x) of (∗), (∗∗) then u = ˜ u by the uniqueness theorem for holomorphic functions.

Definition. Let an entire function F (x) be given by F (x) = X

|α|≥0

f (α) α ! x

^α

,

where α ! = α

1

! . . . α

n

! A function ˇ F (x) is called the Borel transform of F (x) if F (x) = ˇ X

|α|≥0

f (α)

ξ

^α+I

, where I = (1, . . . , 1).

We will need the following result ([4]): if F (x) is an entire function of exponential type σ = (σ

1

, . . . , σ

n

) then ˇ F (x) is holomorphic in Γ

σ

= {x ∈ C

ⁿ

: |x

j

| > σ

j

> 0, j = 1, . . . , n}.

Choose r

_j

> σ

_j

; then ˇ F is holomorphic in the closed domain ¯ Γ

_r

= {x ∈ C

ⁿ

: |x

_j

| ≥ r

j

, j = 1, . . . , n}.

Let us denote by γ

ξ

the set {x : C

ⁿ

: |ξ

j

| = r

j

, j = 1, . . . , n}. Under the above assumptions we have the integral formula

(2) F (x) = 1

(2πi)

ⁿ

R

γξ

F (ξ) e ˇ

^xξ

dξ,

where xξ = x

1

ξ

1

+ . . . + x

n

ξ

n

, dξ = dξ

1

∧ . . . ∧ dξ

n

.

(3)

CAUCHY PROBLEM IN A CLASS OF ENTIRE FUNCTIONS

191 3. We can now formulate our main result. We choose r = (r

1

, . . . , r

n

) so that the functions ˇ v

k

(x) are holomorphic in ¯ Γ

r

for k = 0, 1, . . . , m−1. Then we choose r

0

satisfying the following conditions:

(i) solutions of P (τ, ξ) = 0 belong to {τ ∈ C : |τ | < r

0

} for all ξ ∈ γ

ξ

, (ii) the function ˇ Φ(t, x) is holomorphic in ¯ Γ

r0,r

⊂ C

ⁿ⁺¹t,x

Definition. Let K(ˇ v(ξ), b(ξ), τ ) denote the function

(3) K(ˇ v, b, τ ) =

m−1

X

k=0

X

µ+ν=k

ˇ

v

µ

(ξ) b

ν

(ξ)

τ

^m−k−1

.

Theorem. A solution of the problem (∗), (∗∗) is given by the formula

(4) u(t, x) = 1

(2πi)

ⁿ⁺¹

R

γ_τ×γ_ξ

[K(ˇ v, b, τ ) + ˇ Φ(τ, ξ)] e

^{tτ +xξ}

dτ ∧ dξ

P (τ, ξ) .

P r o o f. Substituting (4) into (∗) we obtain P (D)u = I

1

+ I

2

, where I

1

= 1

(2πi)

ⁿ⁺¹

R

γ_τ×γξ

K(ˇ v, b, τ ) e

^{tτ +xξ}

dτ ∧ dξ,

I

₂

= 1 (2πi)

ⁿ⁺¹

R

γ_τ×γξ

Φ(τ, ξ) e ˇ

^{tτ +xξ}

dτ ∧ dξ.

Since K(ˇ v, b, τ ) is holomorphic with respect to τ for all ξ ∈ γ

ξ

we have I

1

= 0. From (2) we conclude I

2

= Φ(t, x). It follows that P (D)u = Φ(t, x). We only need to show that D

^k

u = v

k

(x) for t = 0, k = 0, 1, . . . , m − 1. Substituting (4) in (∗) we obtain D

_t^k

u|

t=0

= I

3

+ I

4

, where

I

₃

= R

γ_ξ

R

γ_r

K(ˇ v, b, τ ) τ

^k

dτ P (τ, ξ)

e

^xξ

dξ,

I

4

= R

γ_ξ

R

γ_r

Φ(τ, ξ) τ ˇ

^k

dτ P (τ, ξ)

e

^xξ

dξ.

Expanding K/P , ˇ Φ/P in powers of 1/τ we conclude that D

^k_t

u|

_t=0

= v

_k

(x) (note that we have used (2) again).

4. Example. Let P (τ, ξ) be a homogeneous polynomial in two complex variables (τ, ξ). We will denote by λ

_j

the roots of P (λ, 1) = 0. For simplicity we assume that λ

_j

are simple roots. According to the above remark, we have

P (τ, ξ) =

m

Y

j=1

(τ − λ

j

ξ),

b

_k

(ξ) = (−1)

^k

X

1≤i₁<...<i_k≤m

λ

_i₁

. . . λ

_i_k

ξ

^k

= σ

_k

(λ) ξ

^k

, k = 1, . . . , m,

b

0

(ξ) = 1.

(4)

192

E. LEINARTAS

We define P

_µ

(v) = P

₁

(P

_µ−1

(v)) and P

₁

(v) = R

ξ

0

v(ξ) dξ. Under the above assumptions and notations we have

(5) u(t, x) =

m

X

j=1

1 ( Q

m

k=1 k6=j

(λ

_k

− λ

j

))

m−1

X

i=0

X

µ+ν=i

P

µ

(v

µ

(x + λ

j

t)) σ

ν

(λ)

λ

^m−i−1_j

. P r o o f. We first compute (4) by the residue theorem in variable τ , next we use the following generalization of (2):

R

γ

ξ

^µ

F (ξ) e ˇ

^{x ξ}

dξ = P

µ

(F (x)).

In particular, for the wave equation

∂

²

u

∂t

²

= a

²

∂

²

u

∂x

²

we have m = 2, λ

1

= a, λ

2

= −a; then

u(t, x) = 1

2 [v

0

(x + at) + v

0

(x − at)] + 1

2a [P

1

(v

1

(x + at)) − P

1

(v

1

(x − at))].

This is d’Alembert’s well known formula.

References

[1] M. M i y a k e, Global and local Goursat problems in the class of holomorphic or partially holomorphic functions, J. Differential Equations 39 (1981), 445–463.

[2] J. P e r s s o n, On the global and local non-characteristic Cauchy problems when the solutions are holomorphic functions or analytic functionals in space variables, Ark. Mat. 9 (1971), 171–180.

[3] B. Y. S t e r n i n and V. E. S h a t a l o v, On some integral transformations of complex-analytic functions, Dokl. Akad. Nauk SSSR 280 (1985), 553–556.

[4] L. I. R o n k i n, Introduction to the Theory of Entire Functions in Several Variables, Nauka,

Moscow, 1971.

ON THE CAUCHY PROBLEM IN A CLASS OF ENTIRE FUNCTIONS