A refinement of Kovalevskaya’s theorem on analytic solvability of the Cauchy problem Текст научной статьи по специальности «Математика»

УДК 517.53+517.55

A Refinement of Kovalevskaya's Theorem on Analytic Solvability of the Cauchy Problem

Alexander A. Znamenskiy*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 25.11.2016, received in revised form 20.05.2017, accepted 10.07.2017 In this paper we give a proof of an analog of the Kovalevskaya theorem about analytic solvability of the Cauchy problem for a linear differential equation with constant coefficients. A major role in the proof is played by the Borel transform and the Laurent expansion of the function P, where P is the characteristic polynomial. This expansion produces an efficiently computable approximation of the solution of the Cauchy problem. The method of the proof allows to consider equations not necessarily resolved with respect to the highest derivative, however it imposes additional restrictions on the right hand side.

Keywords: Cauchy problem, Borel transform, Newton polytope, Laurent expansion. DOI: 10.17516/1997-1397-2017-10-4-531-536.

Introduction

Consider a Cauchy problem for a linear partial differential equation. In the traditional formulation of the Cauchy-Kovalevskaya theorem it is assumed that the equation is resolved with respect to the pure (not mixed) derivative of the highest order, for example, with respect to dm/dxm, where m is the order of the differential equation. Namely, the equations considered are of the form

dmy ,__ /

y J2 aa(x)Day + f, (1)

dx^

\ a\ ^m

where

t(x) =

Xxi,... ,xn), Day =

dai

dxa ■ ... ■ dxan :

dmy dxm'

(2)

and the summation is taken over derivatives of orders |a| := ai + • • • + an < m, except for In this case the initial data is usually the following

dk y , ,

dxk(x , 0) = yk(x ), k = 0, ..., m - 1,

where x' = (x1,..., xn-1).

Kovalevskaya proved [1] that for any analytic in a neighborhood of the origin coefficients, aa(x), function f (x), and initial data (2) the problem (1), (2) has a unique analytic solution y(x).

In the case of constant coefficients we consider a more general equation than (1). Namely, let P be a polynomial

P

+ E(

aeA

(3)

* [email protected] © Siberian Federal University. All rights reserved

a.

y

a

ai ...a

a.

a

n

where A c Z"0 x {0,1,... ,m — 1} is a fixed finite set of exponents. Then P as a characteristic polynomial defines a differential operator P(D); we consider a differential equation

P (D)y = f (4)

with right hand side f = bkxk given by a power series.

Note that the leading monomial of P with respect to variable zn has the form z^, the degree of zn in all other monomials is less than m, degrees of the remaining variables are arbitrary. For

dm

equation (4) this implies that it contains the derivative d—, which is the highest with respect

n

to xn, but not necessarily the highest derivative in the equation. Then the Cauchy problem for equation (4) consists in finding a solution y(x) of (4) with initial data of the form (2).

Note that we can always assume that the initial data vanish making the substitution y = y+$, where

m— 1

$ =Y xnyj(x'). (5)

/ j j! j=0 J•

Then after substitution of y + $ instead of y in the k-th equation of the system (2), k = 0, 1, . . . , m — 1 we get the equality

—y + -dL Y j) = yk {X')

dxn dxn j=0 j!

rm 1 xj y ■ ( x')

After differentiation of the sum ^ n j- the first k — 1 summands will be zeroes, the k-th

3 = 0 j!

summand will become yk(x'), and the rest will vanish since they contain the vanishing factor xn. Moving yk(x') to the left hand side of the equality and simplifying the expression, we get zero initial data

d k

d^y = 0, k = 0,1,... ,m — 1. (6)

Since this transformation does not change the form of the equation, without loss of generality we assume that we are given zero initial data from the beginning.

Now we can formulate the theorem that will be proved in the rest of the paper.

Theorem 1. If the right hand side f of equation (4) is an entire function of exponential type then the Cauchy problem (4), (2) has a unique analytic solution.

The severe restriction on the right hand side f of the equation is dictated by the Borel transform used in the proof. The condition of f being an entire function of exponential type ensures that the Borel transform of f has a non-empty domain of convergence.

Note that for an arbitrary support A of summation in (3) the restrictions on the right hand side of the equation become essential, which is demonstrated by the well-known example of Kovalevskaya for the heat equation ([1], p. 22).

Thus, a more general form of the equation implies stricter conditions on the right hand side. And vice versa, if an equation is in the generalized Kovalevskaya class then, as Korobeinik showed [2], the existence of the solution is established even for more general classes of functions than analytic.

Note that in relation to equations with constant coefficients of the form (1) the Borel transform has been employed in [3] to obtain an integral representation for a solution of the corresponding Cauchy problem.

1. The Borel transform and properties of the characteristic polynomial

Definition 1. Let the function

f (x)= Y, bkxk

kez^0

be analytic in a neighborhood of the origin. The function

BfM = £ 'Zk+r,

kez™0

where I is the unit multi-index, is called the Borel transform of f [4].

Let us assume that the Taylor coefficients bk of f decrease rapidly to ensure that the domain of convergense of the series Bf is not empty. For example, entire functions of exponential type have this property. By Abel's lemma such a domain of convergence must contain the set lz1l > R1,..., lznl > Rn. Therefore the series converges uniformly on the torus lz1l = R1,..., lznl = Rn, and by term-wise integration we get

J . J Bf (z)e<z,x) dz1 A ... A dzn = f (x). (7)

\z1\ = R1 \zn\ = Rn

Let A be a polytope, i.e. the convex hull of a finite set of points in Rn. Definition 2. The dual cone of the polytope A at the point p € A is the set

Cp = Cp(A) = {q € (Rnf : max(q, a) = (q,p)}.

aeA

Thus, the dual cone Cp consists of all functionals q whose maximal values on A are attained at the point p [5].

Definition 3. Let P = Y1 aaxa be a polynomial. The Newton polytope AP of P is the

aeA

convex hull in Rn of the set A C Zn C Rn of exponents a such that aa is not zero. By C we shall denote the interior of the set C € Rn.

Lemma 1. Let v be a vertex of the Newton polytope AP of the polynomial P, and Cv be the dual cone of AP at the vertex v. Then for any c € Cv the torus

Tp = {lz1l = PC1,..., lznl = Pcn} does not intersect the hypersurface V = {z : P(z) = 0} for p ^ 1. Proof. Let P(z) = aaza. The restriction of P(z) on Tp is

aeA

PItp =J2 aaei{9'a)P{c'a). (8)

aeA

Since c € Cv and v is a vertex of AP, the maximum on AP of the scalar product (c, a) is attained at the only point v. This means that for sufficiently big P the monomial

pM

dominates in absolute value the remaining monomials as well as their sum. Therefore, the restriction (8) does not vanish for p ^ 1. □

Let now P be a polynomial of the form (3). It is obvious that the point v = (0,..., 0,m) is a vertex of AP.

Lemma 2. For v = (0,..., 0,m) there exists c £ Cv such that for sufficiently big p ^ 1 the torus Tp lies in the domain of convergence of the series Bf and does not intersect the hypersurface

{z : P(z) = 0}.

Proof. Let us show that the dual cone Cv (AP) at the vertex v = (0,..., 0,m) contains a vector c = (c1,..., cn) with positive coordinates. The vector c we shall find in the form (e,...,e, 1), where e is real positive.

The fact of c lying in the interior Cv (AP) is expressed by the condition that the maximum

max (c, a) (9)

aeAp

is attained at the only point a = v.

The value of the function (c, a) for a = v is equal to m. For all other points a £ AP

(c, a) = (e(ai + ... + an-i) + an), (10)

and an < m. Since the function a1 +... + an-1 is bounded on the compact set AP, the maximal value of (10) is less than m provided e is sufficiently small. Thus, the existence of c is proved.

Now by Lemma 1, for p ^ 1 the torus Tp = {\z1\ = pci,..., \zn\ = pCn} does not meet the zero set of the polynomial P(z). For sufficiently large p the torus Tp lies within the domain convergence of the series Bf. □

2. The proof of Theorem 1

Introduce the following integral with parameter x:

v(x) = at f i Bf ^ dz1 A ... A dzn := <f B ^ dz. (11)

' (2ni)n J J P(z) 1 n J P(z) [ J

\zi\ = Ri \zn\ = Rn

Note that for Rj = pCj, where p ^ 1, this integral is well-defined: according to Lemma 2 in this case the set of integration does not intersect the zeroes of the integrand's denominator. Differentiating the integral, we see that (11) satisfies equation (4):

P(D)y(x) =f Bf {z)Ppdz = £ Bf (z)e<z•x dz = f (x).

Here we use the equality P(V)e''z'x = P(z)elz'x.

Represent the polynomial P(z) as P(z) = zm — 0(z). Then

P (z) zm — 8(z) zm(1 — zm)

n v-1- zm ' n l = 0

1 = 1 = 1 = 1 V (0-(z±\l-

zm(1 — m.) = zm f=0\ zm J'

nn

this series converges since the monomial zm on Tp dominates in absolute value the sum of the remaining monomials, as follows from the proof of Lemma 1. It follows that the integral (11) is given by the following series

y(x) = iJf(z)eMdzud-mBf(z)eizxdzud-m0MBf(z)eizxdz+.... (12)

vo(x) vi(x) V2(x)

Let us show that the solution y(x) satisfies homogeneous initial data (6). In order to do this it is enough to show that each term of the sequence vv satisfies the conditions (6). We have

£ * (f)" ^^ dz

dzn J \ zn

j J (zT+1' ■ zkn+1+m-s

zm I e ' Z.^ (z')k'+I' . zn + 1+m-s n 7 J

bk ■ k !

Here the degree of zn in each summand-1—-n-does not exceed -2, since s < m,

O n (z')k' + I' ■ zn+1+m-s )

i 0( z)\ u

k ^ 0 and therefore k + 1 + m — s ^ 2 . The degree of zn in the expression i —m j can not be positive. Taking into account that

f dz = jl, if j = I, J zj lo, otherwise,

we obtain

ds

——vv(0)=0, m - l > s > O.

dzS

n

Now we construct a sequence of functions convergent to a solution of (4). Denote by D-mf an antiderivative of f of order m with respect to variable xn. Such antiderivatives are defined up to polynomials of degree m — 1. For instance, if a function f (x) is given by a series

f (x) = ]T bkxk, then its antiderivative D-mf (x) is given by the series

oo

b

^ ^ (kn~+ m) ! v*~ ' ~n

e E (f^m (x')n<n+m

kn = -m 'keZZ

n

Z>0

We shall only choose antiderivatives such that bk = 0 for all kn < 0. This implies that the solution satisfies the homogeneous initial data. Thus, we have a sequence

vo(x) = D-mf,

v1(x) = (V-md(V)) D-mf,

Vv (x) = (V-me(V)Y D-mf,

For the partial sum yv (x) = ^ vk (x) we obtain

k=0

vmyv (x) = Dm (D-mf + D-me D-mf + ■■■ + (D-me)v D-mf) = f (x) + e (D) y^x).

Passing to the limit as v ^ œ, we get Dmy(x) — e (D) y(x) = f (x).

It is necessary to note that this iteration coincides with the iteration from a well-known theorem of Hormander [6]. However its convergense is established there differently. Using this iteration we can consider (as a series (12)) equation (4) with variable coefficients (and even some non-linear equations).

References

[1] Sophie von Kowalevsky, Zur Theorie der partiellen Differentialgleiehungen, Journal fur Mathematik, 1874, Bd. LXXX. Heft 1.

[2] Yu.F.Korobeinik, Representative systems of exponentials and the Cauchy problem for partial differential equations with constant coefficients, Izvestiya: Mathematics, 61(1997), no. 3, 553-592.

[3] E.Leinartas, The Cauchy problem in a class of entire functions in several variables, Banach Center Publications, 33(1996), no. 1, 189-192.

[4] L.I.Ronkin, Introduction to the theory of entire functions of several variables. Translations of Mathematical Monographs, 1974.

[5] A.G.Khovanskii, Newton polyhedra and toroidal varieties, Funct. Anal. Appl., 11(1977), no. 4, 289-296

[6] L.Hormander, The Analysis of Linear Partial Differential Operators II. Differential Operators with Constant Coefficients, 2005.

Одно уточнение теоремы Ковалевской об аналитической разрешимости задачи Коши

Александр А. Знаменский

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

В статье приводится доказательство аналога теоремы Ковалевской об аналитической разрешимости задачи Коши для линейного дифференциального уравнения с постоянными коэффициентами. В этом доказательстве важную роль играют преобразование Бореля и разложение Лорана функции P-1, где P — характеристический многочлен. Такое разложение продуцирует рационально вычислимую аппроксимацию решения задачи Коши. Этот метод доказательства позволяет 'рассматривать уравнения, не обязательно разрешенные относительно производной старшего порядка, однако накладывает ограничение на правую часть уравнения.

Ключевые слова: задача Коши, преобразование Бореля, многогранник Ньютона, разложение Лорана.