Asymptotic Solutions of Second Order Equations with ... · Korovi na M. V., Existence of Resurgent...

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Asymptotic Solutions of Second Order Equations with Holomorphic Coefficients with Degeneracies and Laplace’s Equations on A Manifold with A Cuspidal Singularity

By M. V. Korovina

Abstract- In this paper, we construct the asymptotics for second order linear differential equations with higher-order singularity for the case where the principle symbol has multiple roots. In addition, we solve the problem of constructing asymptotic solutions of Laplace’s equation on a manifold with a second order cuspidal singularity.

Keywords: differential equations with cuspidal, singularitus, laplas-borel transformation, resurgent

function, laplace’s equation.

GJSFR-F Classification: MSC 2010: 44A10

AsymptoticSolutionsofSecondOrderEquationswithHolomorphicCoefficientswithDegeneraciesandLaplacesEquationsonAManifoldwithACuspidalSingularity

Strictly as per the compliance and regulations of:

Global Journal of Science Frontier Research: FMathematics and Decision SciencesVolume 17 Issue 6 Version 1.0 Year 2017Type : Double Blind Peer Reviewed International Research JournalPublisher: Global Journals Inc. (USA)Online ISSN: 2249-4626 & Print ISSN: 0975-5896

Asymptotic Solutions of Second Order Equations with Holomorphic Coefficients

with Degeneracies and Laplace’s Equations on a Manifold with a Cuspidal Singularity

M. V. Korovina

Abstract-

In this paper, we construct the asymptotics for second order

linear differential equations with higher-order singularity

for the case where the principle symbol

has multiple roots. In addition, we solve the problem of constructing asymptotic solutions of Laplace’s equation on a manifold with a second order cuspidal

singularity.

Keywords:

differential equations with cuspidal, singularitus,

laplas-borel transformation, resurgent function, laplace’s equation.

I.

Introduction

This paper is devoted to asymptotic expansions for solutions to equations with

higher-order degeneracies, namely, to equations of the form

,0,,1, 1 =

∂∂

−− + ux

ixdrdr

krH k (1)

where

l

kj

jjl

l

k

drdr

xirxa

xix

drdrrH

∂∂

−=

∂∂

− +

= =

+ ∑∑ 12

0

2

1

1 ),(,,,

−k integer non-negative number,

),( rxa jl

holomorphic

coefficients

in the neighbourhood of zero in

variable r.

Here Cr∈

and x

belongs to a compact manifold without

edge. Such equations are referred to as equations with cuspidal singularitus

of order k+1,

for

0=k , such singularitus are said to be conical. The case of conical singularitus was studied dy Kondratev in [1]. Here we

consider the

case of cuspidal singularitus

. Note that any linear differential equations of

second order

with holomorphic coefficients with

singularitus in

one of the variables

is

representable in the form (1).

Laplace’s equation on a manifold with

cuspidal singularity

is a typical example of such an equation.

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© 2017 Global Journals Inc. (US)

Author: e-mail: [email protected]

In the first part of the article we construct asymptotic solutions of ordinary differential equations of second order with coefficients ( ) 2,1,0, =irai which isholomorphic in some neighborhood of the point 0=r

1.K

ondra

tev V

.A.

Boundary

Valu

e P

roble

ms

for

Ellip

tic

Equations

in D

omain

s w

ith

Con

ical

or

Angu

lar

Poi

nts

, T

r. M

osk. M

at. O

bs.

,196

7, v

ol,1

6, p

p 2

09-2

92.

Ref

( ) ( ) ( ) ( ) ( ) ( ) .001

2

2 =+

+

ruraru

drdraru

drdra (2)

( ) 2,1,0, =irai

are is holomorphic in some neighborhood of the point 0=r

In paper

[5] received asymptotic solutions

of equation

0, 2 =

− u

drdrrH , (3)

where ( ) ( ) in

ii praprH ∑

=

=0

, ,

with the roots of principle operator symbol ( )pHpH ,0)(0 =

being

simple. The asymptotic solutions of

the above equation have the form

∑ ∑=

∞

=

=n

i k

kki

r rareu ii

1 0

/ σα (4)

where

nii ,...,1, =α -are

the roots of ( )pH ,0

and

iσ

and

kia –are

some complex numbers. However, if the asymptotic expansion has at last two terms corresponding to values 1α

and

2α

with distinct real parts (to be definite, we assume that 21 ReRe αα > ),

then it becomes quite difficult to interpret

the rights hand part of (4). The point is that all terms of the first

element corresponding to the value 1α

(the dominant component) have a higher order as 0→r

then any term of the second (the recessive element). If the argument r moves in the complex plane, then

the role of the

components can be changed. Therefore, to interpret the expansion (4), one should sum the (not necessarily convergent)

series (3), the analysis of asymptotic expansions of solutions of equations (1) requires the introduction of regular summation method

for

divergent series for the construction of uniform asympto tic expansions of solutions with respect to the variable r.

In paper

[2] and [5] author examined the

conditions

of

infinite continuable for

Laplace-Borel k-transforms

of

solutions

to

these

equations

and

proved their continuability

along

any path on the Riemann surface not passing through a certain discrete set of points depending on the function, the exact definition of

resurgent function is given in below.

Based on the concept of resurgent function

first introduced by

J. Ecalle

[3], apparatus for summing expressions of the form (4), based on the Borel-Laplace transformation

is called resurgent analysis.

The fundamentals of

resurgent analysis

and of the Borel-Laplace transform

are

based on can be found in

[4].

In articles [7], [8] asymptotic solutions

of equations

( ) ( ) ( ) ( ) ( ) 001

2

=+

+

xuxaxu

dxdxaxu

dxd

are constructed in

the neighborhood of infinity, provided that the coefficients ( )xai

are holomorphic in the neighborhood of infinity.

This equation

is reduced to the

second


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( F)

Asymptotic Solutions of Second Order Equations with Holomorphic Coefficients with Degeneracies and Laplace’s Equations on a Manifold with a Cuspidal Singularity

order equations with cuspidal singularitus in the neighborhood of the point 0=r , by

substituting r

x 1= , which is a private case of tasks which are considering in that paper,

that is equations of any order singularitus.

2.K

orovin

a M. V

., Existen

ce of Resu

rgent S

olution

s for Equaton

s with

High

er –O

rder

Degen

erations, D

iffer. Urav

n, 2011, v

ol. 47,no.3,

pp. 349-357.

Ref

In the second part of the paper we consider

partial differential equations

with cuspidal singularitus.

As an example we construct asymptotic solutions of Laplace’s equation on a manifold with

a second order caspidal singularity.

The paper continues the

research into asymptotic behaviour of solutions

to equations with singularit carried out in a series of articles [2], [5], [6] and so on.

In [5], asymptotic expansions for solutions to equations

of type (1) are constructed for 1=k , and, in [6], they are constructed for 1>k

if the roots of the principle

operator symbol are simple. The problem of asymptotic solutions in the case of multiple roots is much more complicated and is still an open problem. This paper proposes a method for obtaining asymptotic solutions of second order equations with higher-order singularity

in the case of multiple roots. The method is also applicable to some types of higher-order ordinary or partial differential equations.

II.

Basic

Definitions

In this section we introduce

some notions

of

resurgent analysis for further use.

Let ε,RS denote the sector RrrrSR <<<−= ,arg, εεε .

We say that the function f

analytic at ε,RS

has at most k-exponential growth if there exist

nonnegative

constants С

and α

such that

the inequality

kra

Cef1

<

holds in

the

sector ε,RS .

Let ( )ε,Rk SE

denote the space of holomorphic functions of

k-exponential growth,

and let ( )ε,~

RE Ω

denote the space of holomorphic functions of

exponential growth in

ε,~

RΩ .The domain ε,~

RΩ

is shown in Fig1. ( )CE

will denote the space of entire

functions

of

exponential growth.

The Laplace-Borel

k-transform of the function ( )ε,)( Rk SErf ∈

is given

by

∫ +−=

0

01

/ .)(r

krp

k rdrrfefB

k

We can show that ( ) )/E(C)~

E(: ,, εε RRkk SEB Ω→ . The inverse

-transform

is defined by

.)(~2

~~

/1 ∫=−

γπdppfe

ikfB

krpk

where

γ~

is shown in Fig. 1.

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( F)


5.K

orov

ina

M.

V.

and S

hat

alov V

. E

., D

iffe

rential

Equat

ions

with D

egen

erat

ion a

nd

Res

urg

ent

Anal

ysi

s, D

iffe

r. U

ravn, 20

10, vol

. 46

, no

9, p

125

9-12

77.

Ref


Fig. 1: The Laplace-Borel transform domain of holomorphism and the reverse transform calculation

We can now give the definition of a k-resurgent function.

Definition 1. The function f~ is called k-endlessly continuable, if for any R there exists a discrete set of points ZR

in C such that the function

f~ can be analytically continued

from the initial domain along any path of length < R

not passing through ZR.

Definition 2.

The element f

of the space ( )ε,Rk SE

is called a k-resurgent

function, if its

Borel k-transform

fBf k=~

is

endlessly continuable.

III.

Asymptotic

Solutions

of

Ordinary

Differential

Equations

In this section we consider second-order homogeneous ordinary differential equations with cuspidal

singularitus, that is

the equations of the form

0)1,( 1 =+

drdr

nrH n ,

where the symbol ),( prH

is second-order polynomial in p

with holomorphic coefficients.

In [2] it is proved that solutions of these equations are resurgent functions

as formulated

in

Definition 2. Here

we assume that the principle

symbol

),0()(0 pHpH =

has a multiple root. In

the case of

simple roots of a principle

symbol

such equations are discussed in [6]. In other words, we consider the equations of the form

( ) ,01)(1)(1 2

10

11

21 =

++

+

+++ u

drdr

nrrvurau

drdr

nrau

drdr

nnnn (5)

where

)(rai are holomorphic coefficients.

)()(),()(

11

0

11

1

rcrcrrarbrbrra

pp

kk

+

+

+=

+=

Here

)(),( 11 rcrb

are holomorphic functions, with non-zero

с

and

b.

Thus we

rewrite

equation (5) as


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( F)

1)(11 11

112

1 +

++

+

++++ u

drdr

nrbrucru

drdr

nbru

drdr

nnkpnkn

6.K

orovin

a M

. V

. A

sym

ptotic

Solu

tions

for E

quaton

s w

ith H

igher –

Ord

er D

e-gen

erations, D

oklad

y M

athem

atics Vol. 437, N

o. 3, 2011 p. 302-304.

Ref


.01)()(2

11

1 =

++ ++ u

drdr

nrrvurcr np (6)

Since for any 10: +<< nmm

the relation`

2121

21

+

=

+−+−++

drdrr

drdrmr

drdr mnmmnmnn ,

holds, then equation (6)

can be rewritten in the form

01)(

1)()(1)(

111

11

2112

111

11

112

12

=

+

+

++

+

++

+

+

+−++

+−+++−++

+−++−++−

udrdr

nrvr

nm

udrdr

nrvrurcru

drdr

nrbr

ucrudrdr

nbru

drdr

nr

nmu

drdr

nr

mnnm

mnmpmnmk

pmnmkmnmnmnm

(7)

The following two cases are considered.

1. np≥

2and

nk ≥ .

This is the simplest case. We set nm = , then equation (7) has the form

01)(

1)()(1)(

111

12

212

11

11

22

2

=

+

+

++

+

++

+

+

+

++++

+

udrdr

nrvr

udrdr

nrvrurcru

drdr

nrbr

ucrudrdr

nbru

drdr

nru

drdr

nr

n

npnk

pnknn

(8)

Dividing (8) by

nr 2 , we obtain the equation

01)(

1)()(1)(

111

2

112

11

22

=

+

+

++

+

++

+

+

+−+−

−−

udrdr

nrrv

udrdr

nrrvurcru

drdr

nrbr

ucrudrdr

nbru

drdr

nu

drdr

n

npnk

npnk

This is an equation with a conic singularity. In other words, it is not the case of

a cuspidal singularity.

This case is have been studied extensively

(e.g.

[1]).

It is

reasonable to search for

the solutions of these equations in weighted Sobolev spaces

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( F)


( )∞,0,σkH . For nonhomogeneous equations a solution can be represented in the form ofconormal asymptotics. For homogeneous equations, it is identically equal to zero.

1.K

ondra

tev V

.A.

Boundary

Valu

e P

roble

ms

for

Ellip

tic

Equations

in D

omain

s w

ith

Con

ical

or

Angu

lar

Poi

nts

, T

r. M

osk. M

at. O

bs.

,196

7, v

ol,1

6, p

p 2

09-2

92.

Ref


2.

Let

np<

2or nk < , and if

2pk < , then,

similarly

to the conic case,

we set km =

in

(7).

Divide

the equation (7) by kr 2

and multiplying by 2

− knn

we obtain the

equation

( )

01)(1)(

)(1)(

111

21

21

1

2211

12

2

2

112

1

=

−−+

−+

+

−+

−−+

−+

+

−−+

−−+

−

+−+−

−++−−

+−+−−+−

udrdr

knrrv

knnu

drdr

knrrv

urckn

nrudrdr

knknnrrbur

knnc

udrdr

knknnbu

drdr

knr

knnu

drdr

kn

knkn

kpknkp

knknknkn

The principle

symbol in this case is equal to

( ) pkn

nbppH−

+= 20 , and has

simple roots.

In [6],

the solution of this equation is shown to belong to

the space

( )ε,Rkn SE − . Asymptotic

solutions for this equation has

the form

( ) ( ) ( ) .expexp11

11

101

10

10

∑+

−−+

∑ −−

−−−

=−−−

−−−−

=iknikkn

iknikn

knkn

i rrknbnrvr

rrvr αα σσ

In this case ( ) 1,0, =irvi

denotes,

in general,

divergent

series

( ) 1,0,1

==∑∞

=

irvrv j

j

iji

and

( ) 1,0, =iriσ

and

ijα

denote corresponding numbers. The method to calculate them

is outlined in [6].

Now assume that kp<

2

then, we set 2pm = ,

in equation (7) and divide it by pr ,

thus obtain

the equation

01)(2

1)()(1)(

112

1

121

21

21

12

12

1

122

122

21

2

=

−

+

+

++

+

++

+

+

+−−+

+−+−−+

+−−+−−+−

udrdr

qnrvr

np

udrdr

nrrvurcru

drdr

nrbr

cuudrdr

nbru

drdr

nr

npu

drdr

n

pnqn

pnpnpk

pnpkpnpnpn

(9)

If

qp 2=

is an even number, then the equation takes the form


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( F)

1

11

21

12

1

+

−

+

−−

+

+

−−

+

−

+−−

+−−+−

uqn

ncudrdr

qnqnnbr

udrdr

qnr

qnqu

drdr

qn

qnqk

qnqnqn

6.K

orovin

a M

. V

. A

sym

ptotic

Solu

tions

for E

quaton

s w

ith H

igher –

Ord

er D

e-gen

erations, D

oklad

y M

athem

atics Vol. 437, N

o. 3, 2011 p. 302-304.

Ref


01)(

1)(

)(1)(

11

21

1

21

11

=

−−

+

+

−

+

+

−

+

−−

+

+−+−

+−

+−−+

udrdr

qnrvr

qnq

udrdr

qnrrv

urcqn

nrudrdr

qnqnnrbr

qnqn

qn

qnqk

Then the symbol is equal to ( ) cqn

nppH2

20

−

+= . Its roots are simple.

The

solution of this equation belongs to the space ( )ε,Rqn SE − . In this

case, as shown in the

article [6],

the asymptotics

has the form

( ) ( ) .expexp21

10

11

11

21

∑+−+

∑+−− −−

−−−−

=−−−

−−−−

=− iqn

iqnqn

iqniqn

iqnqn

iqn rr

cqn

nirvr

rr

cqn

nirvr αα σσ

If р

is odd, we make the change

(of)

21

rx =

and by substituting

= +−+−

dxdux

drdr pn

pn 1212

21

into equation (9) we obtain

02

1)(2

2

21)()(

22

21

22)(

22

21

22

21

221

212222

212222

1

22

1221

222

1221222

12

=

−−

+

+

−

+

−

+

+

−−

+

−

+

+

−−

+

−−

+

−

−+−+

−+

−+−+

−+−−+−−+

udxdx

pnxvx

pnp

udxdx

pnxvxuxc

pnnx

udxdx

pnpnnxbxcu

pnn

udxdx

pnpnnbxu

dxdx

pnx

pnpu

dxdx

pn

pnpn

pn

pnpk

pnpkpnpnpn

In this case the principle

symbol is equal to ( ) cpn

nppH2

20 2

2

−

+= .

The

asymptotic solution

of the last equation has the form

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( F)


( ) ( ) ( ) ( ) =

∑+

−+

∑+

−−

−−

=−−−

−−

=−−− 1

12

12

2

20

12

12

1

221

22exp

22exp xvx

xxpncixvx

xxpnci pn

iipn

iipn

pn

iipn

ipn

σσ απαπ

( ) ( )

∑+

−+

∑+

−−=

−−

= −−−

−−

= −−−

21

12

12

122

2

2

21

02

12

122

1

2

21

2

2exp2

2exp rvrrrpn

cirvrrrpn

ci pn

iipn

ipn

pn

iipn

ipn

σσ απαπ

6.K

orov

ina

M.

V.

Asy

mpto

tic

Sol

ution

s fo

r E

quat

ons

with H

igher

–

Ord

er D

e-ge

ner

atio

ns,

D

okla

dy M

athem

atic

s

Vol

. 43

7, N

o. 3

, 20

11 p

. 30

2-30

4.

Ref


Since

i

ii

i

ii

i

ii

i

ii

i

ii rvrrvrvrvrvrv ∑∑∑∑∑

∞

=

+

∞

=

+∞

=+

∞

=

∞

=

+=+==

012

121

12

121

0

112

1

12

2

1

121

1 ,

then the asymptotic solution can be rewritten in the form

( )( ) ( )

( )( ) ( ) .~~

2

2exp

~~

2

2exp

22

21

12

212

122

2

2

21

21

11

212

122

1

2

2

1

+

∑+

−+

+

+

∑+

−−

−−

= −−−

−−

= −−−

rvrrvrrrpn

ci

rvrrvrrrpn

ci

pn

iipn

ipn

pn

iipn

ipn

σ

σ

απ

απ

Here .2,1,~,~1

122

12

1 === ∑∑∞

=+

∞

=

jrvvrvv i

i

jij

i

i

jij

In this case we have obtained a new

asymptotic type, namely the asymptotics with nonintegral degrees r in their exponents.

Let us consider the last case. Suppose

npk <=2

. We set

2pkm ==

and dividing

by mr 2

we obtain

and multiplying on

2

− knn

then

01)(1)(

)(1)(

111

112

1

1

21

1

211

21

=

−−+

−+

+

−+

−−+

+

−+

−−+

−−+

−

+−+−+−

+−

+−+−−+−

udrdr

knrvr

knnu

drdr

knrrv

urckn

nrudrdr

knknnrrb

cukn

nudrdr

knknnbu

drdr

knr

knnu

drdr

kn

knknkn

kn

knknknkn

The principle

symbol in this case has the form ( )2

20

−+

−+=

knncp

knnbppH .

If cb 2≠ , then the polynomial 2

2

−+

−+

knncp

knnbp

has two roots 21, cc

and the

asymptotics has the form


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( F)

( ) ( )

∑++

∑+ −−

−−

=−−−

−−

=− ikn

ikn

iknikn

ikn

ikn rr

crvrrr

crvr11

1

21

01

1

10 expexp 10

αα σσ

If cb 2= , namely2

2

−+

−+

knncp

knnbp , then by replacing ueu r

ckn

n−=~ we

obtain the equation with a multiple root at zero. The asymptotic solution of this equation is constructed in the same way as is shown above and depends on the degreesof degeneracy of the functions )(),( 11 rcrb .

Thus we have cuspidal degeneracies in the case where np<

2or nk < . The

obtained results are written in the table

Notes


Theorem 1:

Let np<

2

or

nk <

then

Conditions

Space

Asymptotics

2pk <

( )ε,Rkn SEu −∈

( )

( ) ( )

∑+

−−+

+

∑

−−

−−−

=−

−−

−−−−

=

iknikkn

ikn

iknknkn

i

rknrbnrvr

rrvr

11

11

101

10

exp

exp

1

0

α

α

σ

σ

Nqqp

pk

∈=

>

,2

,2

( )ε,Rqn SEu −∈

( )

( )

∑+−+

+

∑+−−

−−

=−−

−−

−

−−

=−−

−−

−

1

1

2

1

1

1

1

0

exp

exp

2

1

qn

iiqniqn

qn

qn

iiqniqn

qn

rr

cqn

nirvr

rr

cqn

nirvr

α

α

σ

σ

2pk > ,

p

is odd

( )ε,12

RpnSEu

−−∈

( )( ) (

( )( )

+

∑+

−+

+

∑+

−−

−−

= −−−

−−

= −−−

vrrvrrrpn

ci

vrrvrrrpn

ci

pn

iipn

ipn

pn

iipn

ipn

22

21

12

212

122

2

2

21

21

11

212

122

1

2

~~

2

2exp

~~

2

2exp

2

1

σ

σ

απ

απ

2pk =

and

cb 2≠

( )ε,1 Rkn SEu −−∈

( )

( )

∑++

+

∑+

−−

−−−

=−

−−

−−−−

=−

iknikkn

ikn

iknknkn

ikn

rrcrvr

rrcrvr

11

2

21

101

10

exp

exp

1

0

α

α

σ

σ

Now we proceed to examine

the

higher-order equations

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( F)


Notes


( ) 0)(1)(...110

11

11

11 =+

++

+

+

−+

−+ urbu

drdr

nrbu

drdr

nrbu

drdr

nn

kn

k

kn

where ∑

∞

=

−==imj

jjii kircrb 1,...,0,)(

are entire functions. We assume that 0≠im

ic . The

number im

will be called a degree of degeneracy of

the coefficient )(rbi . Suppose that zero is the root of this equation principle symbol

and the degree of degeneracy of the

coefficients )(1 rb

or )(0 rb

is

no more

than the degree of degeneracy for

any of

the

coefficients .1,...,2),( −= kirbi

The above method is also applicable to this case.

Specifically, the equation is to be divided by pr , where ( )10 ,min mmp = , then we have

the equation with a symbol of the form

−

drdrrH k

pn, , which is solved similarly to the

previous one.

IV.

Partial

Differential

Equations

We consider a second-order partial differential equation with holomorphic

coefficients. It can be represented

in

the form

,0,,1, 1 =

∂∂

−− + ux

ixdrdr

krH k

where x

varies on some compact manifold without boundary. These equations

can be interpreted as

equations with respect to the functions with values

in Banach spaces,

namely

,01,ˆ 1 =

− + u

drdr

krH k

(10)

where

( ) ( )2,1, ,,:ˆ BSEBSEH RkRk εε → .

Here 2,1, =iBi

denote some Banach

spaces

(e.g.

the space ( )ΩsH ).

The degree k of the

function

u

grows exponentially for

0→r ,

for

fixed r

and

p

( ) 21:,ˆ BBprH →

is a bounded operator acting in Banach

spaces

which

is

polynomial

dependent on p

and holomorphic in the neighbourhood of zero.

In what follows, we will assume that the operator family ( ) ( )pHpH ,0ˆˆ

0 =

is a

Fredholm

family. This implies that the operator ( )pH0ˆ

is a Fredholm operator for each

fixed p ,

and there exists Cp ∈0

such that the operator ( )00ˆ pH

is invertible.

In addition to the requirement of the Fredholm property for the family ( )pH0ˆ

we assume that there exists a

cone in C containing the imaginary axis and does not


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( F)

contain the points of the spectrum of ( )pH 0ˆ for a sufficiently large p . The spectrum of

a Fredholm family is a set of points Cp∈ such that ( )pH0ˆ is irreversible.

Suppose that the point ( )pHspecp 01ˆ∈ is such that the following conditions hold :

Notes


1.

The operator-valued function ( )pH 01ˆ −

has the second-order pole at the point 1p .

2.

The dimensionality of the )(ˆker 10 pH

is equal to one.

In this case theorem 1 is also true.

The proof is analogous to that of Theorem 9

in

[5] In this case

the coefficients of the series j

j

ijrv∑

∞

=1

are the elements of

the space 1B .

V.

Laplace’s

Equation

on a

Manifold

with a

Caspidal Singularity

We consider the Laplace equation 0=∆u

on the 2D-Riemannian manifold with

a second order caspidal

singularity. The Riemannian metric induced from

3R

with the help of embedding which is defined as

the surface

by the rotational

of the parabolic branch

2ry =

around axis r0 . in 3R . We choose local coordinates ϕ,r

on the manifold

in the neighbourhood

of zero.

The

metrics

on this manifold is given

by

( ) 2422242222 414 ϕϕ drdrrdrdrrdrds ++=++= ,

hence

( )

∂∂

∂∂

+==

ϕu

rru

rAAgradu 2221

1,14

1, .

and

( ) ( ) ( ) .114

11221

2

22211221

Ar

Arrrr

AhAhrhh

divuϕϕ ∂∂

+∂∂

+=

∂∂

+∂∂

=

Finally, for the Laplace operator we obtain

urr

ur

rrrr

divgraduu2

42

2

22

114

114

1

∂∂

+

∂∂

+∂∂

+==∆

ϕ.

Thus Laplace’s

equation on a manifold has

the form

( ) 0114122

141ˆ

2

2

422

2

2

2

2 =∂∂

+∂∂

+

++

∂∂

+=

ϕu

rru

rr

rru

ruH

. (11)

In other words

( )( ) ( )( )12

,1, ,,:ˆ SHSESHSEH sRk

sRk

−→ εε ,

We rewrite equation (11) in the form

( )

∂∂

++

∂∂

−=∂∂

+

rur

rruruu

drdr 2

2

3

2

22

2

222

1444

ϕϕ (12)

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( F)


Obviously this case does not satisfy condition 2 who formulated in part 3, (see of the paper) Therefore, the results formulated there is not applicable. We perform the Laplace-Borel transform

5.K

orov

ina

M.

V.

and S

hat

alov V

. E

., D

iffe

rential

Equat

ions

with D

egen

erat

ion a

nd

Res

urg

ent

Anal

ysi

s, D

iffe

r. U

ravn, 20

10, vol

. 46

, no

9, p

125

9-12

77.

Ref


( ) ),,(~~4~~~ 2

2

22

13

22 ϕ

ϕϕpfuBru

rrraBruup +

∂∂

−

∂∂

−=

∂∂

+ (13)

where

( ) ( )14 21+

=r

rra , and ),(~ ϕpf

denotes a holomorphic function.

The principle symbol

of this operator is equal

22),(

∂∂

+=ϕ

ppoH .

Therefore

the corresponding operator family

( )pH0ˆ

is a Fredholm one.

The existence of resurgent solution

of this equation follows from the results of

[2], [3].

We denote 1ˆ −= BBrr nn ,

as was done in [5].

If a(r) is

holomorphic in the neighbourhood of r

= 0,

vanishing

at the

origin,

and

∑∞

=

=1

)(k

kk rara

(14)

is its Taylor

expansion, then the function

( ) ( )∑∞

=

−

−−=

1

1

!11)(~

k

k

kk

kpapa (15)

is called a formal Laplace-Borel transform of (14), then as shown in the article [5] the equation

( ) ( ) ( ) pdpfppafBraBp

p′′′−= ∫− ~~~

0

1

(16)

Equality

(16) implies that the right-hand part

of equation (13)

can be transformed

( )

( ) ( ) ( ) ),(~,~ˆ4,~~ˆ

),(~~4~

22

22

22

22

112

0

ϕϕϕ

ϕ

ϕϕ

pfpurpdpupppar

pfuBrur

rrBraBBr

p

p+′

∂∂

−′′′′−−=

=+∂∂

−

∂∂

−

∫

−

Equation (13) is ultimately transformed into following

( ) ( ) ( ) ),(~,~ˆ4,~~ˆ~~

2

222

22

0

ϕϕϕ

ϕϕ

pfpurpdpuppparuupp

p+

∂∂

−′′′′−−=

∂∂

+ ∫ (17)

Decompose functions u~ and f~

into a series on eigen function of operator 2

2

ϕ∂∂


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( F)

( ) ( )

( ) ϕ

ϕ

ϕ

ϕ

ik

kk

ik

kk

epapf

epApu

∑

∑∞

=

∞

−∞=

=

=

1),(~

,~

(18)

Substituting (18) into equation, (17) we obtain

3.J. E

calle. Cin

q ap

plication

s des fon

ctions resu

rgentes, 1984. P

reprin

t 84T 62, O

rsay.

Ref


( ) ( ) ( ) ( ) ϕϕϕ ik

kk

ik

kk

ik

kk

p

pepaepAkrpdepAppar ∑∑∑∫

∞

=

∞

−∞=

∞

−∞=

++′′′−−=1

223 ˆ4~ˆ0

This equality is equivalent to the system of equations

( ) ( ) ( ) ( ) ( ) )(ˆ4~ˆ 22222

0

papAkrpdpAppparpAkp kkk

p

pk ++′′′′−−=− ∫ ,

Here

k

is an integer.

The

cases

0=k

and

0≠k

should be considered separately.

In the first case the singular point of the function ( )pA0

is 0=p , being a pole of second

order. We solve this equation by

the method of

successive approximations.

( ) ( ) ( ) ( )pap

pdpAppparp

pAp

p 0202

201

2~ˆ2

0

+′′

′′−−= ∫

(19)

Substitute ( )pap

pA 0201)( =

into the integral in right-hand

part side of (19).

Applying successive approximation method. Obtain the equality

( ) ( ) pdp

ppGrp

pdpap

pparp

p

p

p

p′

′′

=′′′

′− ∫∫),(ˆ11~ˆ1

00

220

22

where ( ) ( )pappappG ′′−=′ 0~),( .

We represent

this function as

the series

∑∞

=

′=′0

)(),(i

ii pplppG , then similarly to [5] we obtain

( )

( )pgppCpCp

C

dpdppnppbp

pdp

pplr

p

i

ii

p p

pi

iip

p

+++=

=+=′′

′

∑

∫ ∫∑

∫

∞

=−

∞

=

lnln1

)(ln)(1)(

ˆ1

101

210 1111202

22 2

00

where ( )pg

denotes the function holomorphic in the zero neighbourhood, and iC

denotes corresponding constants. The proof of convergence of the successive approximation method is the same as analogous to

the proof in

[6]. Thus we obtain that

the function ( )pA0

with an accuracy of a holomorphic summand in the neighbourhood of

the point 0=p

has the form

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( F)


( ) ( ) ϕϕ ik

kk

ik

kk epAkepAp ∑∑

∞

=

∞

−∞=

=−1

22

( ) ppMp

Mp

MpA i

ii ln

0

001

22

0 ∑∞

=

−− ++=

Here by 0iM we denote corresponding constants.

Now we consider the second case 0≠k . In this case the points kp = and kp −=are first-order poles of the function ( )pAk . Here we have the equation

5.K

orov

ina

M.

V.

and S

hat

alov

V.

E., D

iffe

rential

Equat

ions

with D

egen

erat

ion a

nd

Res

urg

ent

Anal

ysi

s, D

iffe

r. U

ravn, 20

10, vol

. 46

, no

9, p

125

9-12

77.

Ref


( ) ( ) ( ) ( ) 2222

222

22)(ˆ41~ˆ1

0 kppapAkr

kppdpApppar

kppA k

kk

p

pk −+

−+′′′′−

−−= ∫

Equations of this form have been studied in [5]. The asymptotics of the function

( )pAk

in the neighbourhood of the singular point kp = , with an accuracy of a

holomorphic function,

will take the form

( ) ( ) )ln(0

1 kpkpMkp

MpA j

j

kj

k

k −−+−

= ∑∞

=

−

Here by kiM

we denote corresponding

constants. Thus we have demonstrated

that, with an accuracy of a holomorphic function, the solution of equation (17) has the form

( ) ( ) ( ) ϕϕϕ ik

k

iki

j

kik

kk ekpkpM

kpM

pMepApu ∑ ∑∑

∞

−∞= =

−−∞

−∞=

−−+

−+== )ln(,~

0

12

2

Hence it follows that singular points of the function ( )ϕ,~ pu

lie in any half-plane

0Re >> Ap . In order to construct

the solution of problem (17) we represent the

function ( )ϕ,~ pu

as a sum of two functions ( )ϕ,~ pu−

and

( )ϕ,~ pu+ such that the

singularities of the first one

lie to the left of the

line

Ap >Re , and the singularities of the second one lie

to the right of this

line

Ap >Re , where

ZA∉Re .

Then the solution

can be constructed as the inverse transformation of the function ( )ϕ,~ pu−

(see [5]). To put it differently the equality

( ) ( ) ( )

−−+

−+== ∑ ∑

−∞=

∞

=

−−−−

− kiN

k j

jkj

k

ekpkpMkp

Mp

MBpuBru ϕϕϕ0

12

211 )ln(,~,

holds with an accuracy of an entire function.

Hence it follows that the asymptotic solution of Laplace’s

equation (13) has the form

( ) ∑ ∑−∞=

∞

=

− +=N

k

i

i

ki

kirk

rMeer

Cru0

1, ϕϕ

We should note that the solution depends on the representation

( ) ( ) ( )ϕϕϕ ,~,~,~ pupupu +− += . One solution differs from the other solutions by an operator

kernel.

Thus we have proved

Assertion.

Any solution of

equation

of exponential growth (12)

can be represented as


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( F)

( ) ( )

++=

−− ∑ r

A

jj eOru

rCru ϕϕ ,, 1 ,

where А is an arbitrary positive number, and the sum contains a finite number of summands, each of which corresponding to a point kp = located in the half-plane

Ap −>Re , and ( )ruj

have asymptotic expansions

5.K

orovin

a M. V

. and S

hatalov

V. E

., Differen

tial Equation

s with

Degen

eration an

d

Resu

rgent A

naly

sis, Differ. U

ravn, 2010, v

ol. 46, no 9, p

1259-1277.

Ref


l

l

kl

jjirj

j rMeeru ∑∞

=

=0

),( ϕϕ

In conclusion, I would like to express my deep gratitude to V.E.

Shatalov for permanent discussions throughout the work. This work was supported by the Russian

Foundation for Basic Research (project

14-01-00163-а).

References Références Referencias

1.

Kondratev V.A. Boundary Value Problems for Elliptic Equations in Domains with Conical or Angular Points, Tr. Mosk. Mat. Obs.,1967, vol,16, pp 209-292.

2.

Korovina M. V., Existence of Resurgent Solutions for Equatons with Higher –Order Degenerations, Differ. Uravn, 2011, vol. 47,

no.3,

pp. 349-357.

3.

J. Ecalle. Cinq applications des fonctions resurgentes, 1984. Preprint 84T 62, Orsay.

4.

B. Sternin, V. Shatalov, Borel-Laplace Transform and Asymptotic Theory. In-

troduction to Re¬surgent Analysis. CRC Press, 1996

5.

Korovina M. V. and Shatalov V. E., Differential Equations with Degeneration and Resurgent Analysis, Differ. Uravn, 2010, vol. 46, no 9, p 1259-1277.

6.

Korovina M. V. Asymptotic Solutions for Equatons with Higher –

Order De-generations, Doklady Mathematics Vol. 437, No. 3, 2011 p. 302-304.

7.

Olver F. Asymptotics and Special Functions. Academic Press, 1974.

8.

Coddingthon E. A., Levinson N. Theory of ordinary differential equations. 2010.

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( F)


,

Notes


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