Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains

(1)

Poincaré Problem for Nonlinear Elliptic Equations of

Second Order in Unbounded Domains

Guochun Wen

LMAM, School of Mathematical Sciences, Peking University, Beijing, China Email: wengc@math.pku.edu.cn

Received September 27, 2012; revised November 2, 2012; accepted November 10,2012

ABSTRACT

In [1], I. N. Vekua propose the Poincaré problem for some second order elliptic equations, but it can not be solved. In [2], the authors discussed the boundary value problem for nonlinear elliptic equations of second order in some bounded domains. In this article, the Poincaré boundary value problem for general nonlinear elliptic equations of second order in unbounded multiply connected domains have been completely investigated. We first provide the formulation of the above boundary value problem and corresponding modified well posed-ness. Next we obtain the representation theorem and a priori estimates of solutions for the modified problem. Finally by the above estimates of solutions and the Schauder fixed-point theorem, the solvability results of the above Poincaré problem for the nonlinear elliptic equations of second order can be obtained. The above problem possesses many applications in mechanics and physics and so on.

Keywords: Poincaré Boundary Value Problem; Nonlinear Elliptic Equations; Unbounded Domains

1. Formulation of the Poincaré Boundary

Value Problem

Let D be an



N1



-connected domain including the

0

N

infinite point with the boundary  



j j 



2 ₀

C 

in ,

where  . Without loss of generality, we

assume that D is a circular domain in



1

  

1

z  , where the boundary consists of N1 circles   0 N1



z 1



,



z z r



 _j  _j  _j , 1, , N and . In this

article, the notations are as the same in References [1-8]. We consider the second order equation in the complex form

j z  D













, , , Re

, , ,

zz z zz

zz z

u F z u u u

F Qu A u

G G z u u Q

 



  



  















1 2 3

, , ,

,

, , , ,

, , , 1, 2,3,

z

z z zz

j j z

G z u u A u A Q z u u u

A A z u u j



 

  





w j



1, 2,3

,

z D U 0, 



 , 2,3



(1.1)

satisfying the following conditions.

Condition C. 1) ,



are continuous in for almost every point

and Q A for z D.



, , ,



Q z u w U A z uj , ,

,

u w

,

 j 0 j 1 

z D

2) The above functions are measurable in  for all continuous functions u z

 

,w z

 

in D, and satisfy









,2 0

1

, , , , 1, 2,

, ,

p j

L A z u w D k j w D k

   

 

  

 





0, 2 < 0 ,

p p p p k k₀, ₁



2 ,

U

,2

p

L A₃ z u, , (1.2)

in which are non-negative

constants.

3) The Equation (1.1) satisfies the uniform ellipticity condition, namely for any number u and w, U1,

 the inequality











F z u w U, , , 1 F z u w U, , , 2 q U0 1U2,

z D holds, where q₀

 

1

for almost every point  is

a non-negative constant.

 

4) For any function u z C D

 

1

 

,2

p

w z W D , ,



, ,



G z u w





satisfies the condition

1 2

, , z z , 0 , ,

G z u u B u  B u    



, ,



1, 2



j j z

B B z u u j satisfy the condition in which

,2 , 0 , 1, 2,

p j

L _B D _ k   j k

(1.3)

with a non-negative constant 0.

Now, we formulate the Poincaré boundary value problem as follows.



Problem P. In the domain D, find a solution u z



of Equation (1.1), which is continuously differentiable in

D, and satisfies the boundary condition

 

1 2

1 _,

2

. . Re _z , ,

u _{c z u c z}

i e z u c z u c z z

 

 



 



 _ _ _ (1.4)

 

D

in which  is any unit vector at every point on   ,

 

z cos ,

 

x icos ,



y



,

(2)



known functions satisfying the conditions

 

, 0,



1,



0,





C_   k C c_  k C c_ ₂, k₂, (1.5)

where  

 

0 , 



1 2  1



k₀ k₂

cos c 0

, , are non-negative

constants.

If and ₁ on , where n is the

outward normal vector on

 

,n 0

 

, then Problem P is the Dirichlet boundary value problem (Problem D). If

and a₁ on

 

,n

cos 1 0 , then Problem P is the

Neumann boundary value problem (Problem N), and if , and ₁ on , then Problem P is the regular oblique derivative problem, i.e. the third bound-ary value problem (Problem III or O). Now the direc-tional derivative may be arbitrary, hence the boundary condition is very general.

 

,n

cos 0 c 0

The integer

 

arg π 1 2

K _  z

0, K is called the index of Problem P. When the index 

Problem P may not be solvable, and when the

solution of Problem P is not necessarily unique. Hence we consider the well-posedness of Problem P with modi-fied boundary conditions.

0,

K

 

Problem Q. Find a continuous solution _w z u z,

 

_ of the complex equation











1



2 3

, , , , , ,

Re ,

z z

z

w F z u w w G z u w

1 2 ,

F Qw A w A u A G B w B u

  

 

   



(1.6) satisfying the boundary condition

   

1

 

2

 

Re_ z w z  _ c z u c z h z z

 

, , (1.7)

and the relation

 

_

_

d 0,

j j

j z

z b

z

 

  

 

 



1, ,



d j  N

D

,

0 ,

0,

z K N

K N

K



 _ _



 

2 1

1

2 Re z N

j id w z

u z

z  z z

  







(1.8)

where j are appropriate real constants

such that the function determined by the integral in (1.8) is single-valued in , and the undetermined function

is as stated in

 

h z

 





1

0 0

1

0, ,

, , 1, , ,

0, , 1, , 1

, , 1, , ,

Re ,

j j

j

j j K

m m m m

h z j N K

h z

z j N K N

h z j N

h   h ih z z



    _ 

  

 



  _ _{  }

 

 

 _ _ _

 _

 _

 

 _

 





 



in which h j0,1, , N



h m



1, ,  K 1,K0



0

K

j , m

are unknown real constants to be determined appropri- ately. In addition, for the solution w z

 

is as- sumed to satisfy the point conditions

 

Im ,

1, , 2 1, ,

1, , 1, 0 ,

j j j

a w a b

K N K N

j J

N K N K N



 _{ }

 

 

  



   _{ } _ _ _





(1.9)

where









0

1, , ,

1, , 2 1,

j j

j

a j N

a j N K N K N

 

     

 

 



are distinct points, and b j J_j   0



are all real constants satisfying the conditions

 

3, 0 ,

j

b k j J 

3

k

 

(1.10)

for a non-negative constant .

2. Estimates of Solutions for the Poincaré

Boundary Value Problem

First of all, we give a prior estimate of solutions of Problem Q for (1.6).

Theorem 2.1. Suppose that Condition C holds and ε = 0 in (1.6) and (1.7). Then any solution _w z u z,

 

_ of Problem Q for (1.6) satisfies the estimates

 

,

 

, 1 ,

Cw z DC u z D M k (2.1)

0,2 , 2 ,

p z z

L _w  w D_M k (2.2) in which



0



min ,1 2 p ,

  



0, 0, , , ,0



, 1, 2, j j

M M q p k  K D j







 



1 2 3 0 , , .

k_{   }k k k k _C w D _ __C u D _

   

   

, w z u z

 

Proof. Noting that the solution _ _ of Prob-

lem Q satisfies the equation and boundary conditions



1



3





Re , , in ,

z z

w Qw A w A G z u w D

 

(2.3)

   

2

Re_ z w _ c z h z on ,

 

(2.4)

 

0

Im_ a w aj j  _ b j J uj,  , 1 b , (2.5) according to the method in the proof of Theorem 4.3, Chapter II, [2] or Theorem 2.2.1, [5], we can derive that the solution w z

 

satisfies the estimates

 

, 3 ,

C_w z D  M k (2.6)

0,2 , 4 ,

p z z

L w  w DM k (2.7)

(3)



K D j,



, 3, 4

 

   

,

w z u z

where

0, 0, , ,0

j j

M M q p k 

and

1 2

k k k k3L G Dp , . From (1.8), it follows that

 

,  5 w z D

 

, k3,

C u z D__ _ M C (2.8)

 

0,2 5

p z z

L _u u D, _M C__w z D,  _ k₃,



,



(2.9)

in which M5 M5 p D0 is a non-negative constant.

Moreover, it is easy to see that





 





 



,

, ,

, , .

w D

u D

C u D





 

 

 

 

 

 

 



,2 ,2 1 ,2 2

0

, ,

p p

p

L G D L B D C

L B D C

k C w D

   

   

 

 _ _

 

 _ _ 

(2.10)

Combining (2.6)-(2.10), the estimates (2.1) and (2.2) are obtained.

Theorem 2.2. Let the Equation (1.6) satisfy Condition

C and  in (1.6)-(1.7) be small enough. Then any solu-

tion _ _ of Problem Q for (1.6) satisfies the estimates

 

,

 

, 6 ,

Cw z DC u z D M k (2.11)

0,2 , 0,2 , 7 ,

p z z p z

L _w _ w D__L _u D__M k

   

0

,p k,



(2.12)

_{are as stated in Theorem 2.1,}

here



0, 0, , , ,0



, 6,7. j j

M M q p k  K D j

   

,

w z u z

 

 

Proof. It is easy to see that satisfies

the equation and boundary conditions





1 2 3

Re , ,

z z

w  Qw A wA u A G z D

 

(2.13)

 

1 2

   

Re_ z w z   _ c u c z h z z, ,

 

(2.14)

 

0

Im_ a w aj j  _ b j J uj,  , 1 b.

 

(2.15)

Moreover from (2.6) and (2.7), we have









0

3 0

,2 4 0

, , ,

p z z

C w z D M k k C u D

L w w D M k k C u D

 

 



 

 _ _  _ _ 



     

 _ _ _ _ (2.16)



 

and from (2.8)-(2.10), it follows that



 



 





3 0 5 3

,2 4 0 5 3

, , ,

, , .

p z z

C w z D M k k M C w z D k

L w w D M k k M C w z D k

 

 

 



 _ _  _ _ _ _ 

0



 

     

   

 _ _ (2.17)



If the positive constant  is small enough such that Combining (2.8) and (2.18), we obtain

0 3 5

1k M M 1 2, then the first inequality in (2.17)

implies that

 









0 3 5

0 3 8

1 ,

2 1 .

k M

C w z k

M

k M k M k







 



  

0 3

1 D

k M



  

  _

(2.18)



5



8 6

, ,

1 1 ,

C w z D C u z D M M k M k

 

 

   

   

 

  _ _  (2.19)

which is the estimate (2.11). As for (2.12), it is easily derived from (2.9) and the second inequality in (2.17),

i.e.

 





 









0,2 0,2

4 0 5 3 5 3

4 0 5 8 0 4 7

, ,

1 1 1 .

p z z p z

L w w D L u D

M k k M C w z D k M C w z D k

M k M M k M k M k

 



 



 

    

   

     

  _ _ _ _  _ _

     

 

(2.20)

 

3. Solvability Results of the Poincaré

Boundary Value Problem

ed in Condition C, then the nonlinear mapping G:



We first prove a lemma.

Lemma 3.1. If G z u w



, ,



satisfies the condition stat-

  

p,2

 

C D C D L D

   

, ,

z u z w z



G G

defined by _ _ is continuous and

bounded

   









p B D C u D2,

 

, ,

,2 , , , ,2 1, , ,2

p p

(4)

where p p02.

f. In order t ove that e mapping G:

Proo o pr th

   

p,2

 

C D C D L D

Defined by _z u z w z,

   

, _ is continuous, we

 

 

G G 

choose any sequence of functions w z u z_n

   

, _n

   

 



w z u zn , n C D n, 0,1, 2,



such that

0,  u D0, 0

as .n  Similarly to Lemma 2.2.1, [5], we n prove

0 0

, ,

n G z u w

possesses the property

n n

C w_ w D_C u

ca that



, ,







n n

C G z u w 

,2 ,

p n

L _C D _ 0 asn . (3.2)

And the inequality (3.1) is obviously true.

1.1) satisfy C

Theorem 3.2. Let the complex Equation (

ondition C, and the positive constant  in (1.6) and (1.7) is small enough.

1) When 0, 1, Problem Q for (1.6) has a solution _w

   

z u, z _, where w z

 

,

 

1₀

 

,2

p

u z W D ,



0 p



is a co state .

en

0 2

p p nstant as d before

2) Wh min



 ,



1, Problem Q for (1.6) has a solution _w

   

z u z, _, where

 

1₀

 

,2 ,

p

w z W D pro-

vided that





 

9 ,2 3 2

0

, ,

p j

j J

M L A D C c_ b

 

    

 



(3.3)

is sufficiently small.



3) If F z u w w



, , , _z

 

, , ,



i.e

G z u w satisfy the conditions,

ition C and for any funct

. Cond ions w z_j

 

,

 



1, 2



j

u z C D j and

 

p₀,2

 

D ere are

































1 1 2 2 1 1 2 2 1 2

, , , , , , Re ,

, , , , Re ,

V z L , th

F z u w V F z u w V A w w A u u

G z u w G z u w B w w B u u

 

      

  



 

  _  _ 

 

  (3.4)



where

,2A D L , , p,2B Dj,   k0 , j1, 2,

p j

L

 is a sufficiently small positive constant, th the

above solution of Problem Q is unique.

ation for t is as en

Proof. 1) In this case, the algebraic equ follows







6 7



,2 3 ,2 1 ,2 2 2

{0}

, , , , ,

p p p j

j J

M M L A D L B D t L B D t L a b t

 

     

 _ _ _ _ _  _ _    _





 (3.5)

stated in (2.11) and (2.12).

 

here M6, M7 are constants as

w

Because 0, 1, the Equation (3.5) has a unique

solution 10 Now we introduce a bounded,

closed and convex subset t M 0. B*_{of the Banach space}_{C D}

 

_

 

,

C D whose elements are of the form _w z

   

,u z _

s g the condition

   

atisfyin

 

 

 

 10

(3.6 se a pair of functions w z u z

   

, B

, ,

w z u z C D C_w z D, C u z D , M .

We choo

)





 

  

ositions of

and substitute it into the appropriate p



, , , z



F z u w w , G z u w



, ,



in (1.6) and the boundary condition (1.7), and obtain



, , , , z

w F z u w u w w , z



G z u w



, , 



, (3.7)

   

1

 

2

   

Re z w z _{  }c z u c z_ _h z z,

  

where

,

 (3.8)





















1

2 3

, , , Re , , , , ,

, , , , .

z z z

, ,

F z u w u w w Q z u w w w A z u w w

A z u w u A z u w



 _  _

 

     

   

In accordance with the method in the proof of Theo-re



m 1.2.5, [5], we can prove that the boundary value problem (3.7), (3.8) and (1.6) has a unique solution

   

, w z u z

 

 . Denote by

 

w u,  T w z u z

   

,  the

mapping from _w z u z

   

, _ oting

that

to _w z u z

   

, _. N





,2 2

p A u D,   M k C10 0,  c u1 , M k10 0.

provided that the positive number L

 is sufficiently small, and noting that the coefficients of complex Equa-tion (3.7) satisfy the same condiEqua-tions as in CondiEqua-tion C, from Theorem 2.2, we can obtain

 









 













0,2 0,2

6 7 9 ,2 1 10 ,2 2 1

, , , ,

,

, ,

p z z p z

p

p p

C w z D L w w D C u z D L u D

D

M M M L B D M L B D M

6 7 ,2 3 2 ,2

0

6 7 9 ,2 1 ,2 2

, ,

, , ,

p j

j J

p p

M M L A D C c b L G D

M M M L B D C w D L B D C u



 



        

       

 

 



 

  

  _ _ _    _ _

 

 

      

   _ _{ } _  _ _{ } 



  _

   

 



 _ _  _ _ 0



M10.

(3.9)

(5)

This shows that T maps B*_{onto a compact subset in}_B*_.

Next, we verify that T in B*_{is a continuous operator. In}

fact, we arbitrarily select a sequence



w z u z_n

 

,_n

 



in

B*_{, such that}



n 0,

 

n 0,



C w w D C u u D 0 as n . (3.10) By Lemma 3.1, we can see that











,2 0 0

1, 2,3 as .

p j n n j

j n

 

   (3.11)

Moreover, from

, , ,

L A z u w  A z u ,w ,D0



w u_n, _n

 

T w u _n, _n



,



w u0 0

 

0 0





, T w u , ,

it is clear that n 0 n 0



lution of Problem

equation ,

w w u u is a so

Q for the following

















0

, , , , ,

, , in ,

n z n n n n nz

z

n n

w w F z u w u w w

z u w u w w

G z u w G z D

 

 

 _{ }

 

  

(3.12)





0

0 0 0 0

0

, ,u w F

 

 



 



0

Re z wnw 

 

1 n 0 on ,

c z u u h z



      (3.13)

   

 

0

 

Im

, 1 1

j n j

n

a w a

j J u u



 

 



0



0,

0.

j

w a

 _ _

 

 

 (

In accordance with the method in proof of Theorem 2.2, we can obtain the estimate

3.14)







































 







0

0 ,2 0

11 ,2 2 2 0 0

0 0

1 0

, ,

, , , ,

, ,

n p n z n z

n p n z

p n n n

n

C w w D L w w w

C u u D L u u D

M L A z u w u A z u w

w D

C c z u_ u





    

   

 

 _  _ _  _

 

 _  _



 

 _  _

    

 

(3.15)



0, , ,



.

0

w D



0,

u D

,2 3 3 0 0

,2

, , , , ,

, , ,

p n n

L A z u w A z u w D

L G z u w G z u

 

 _  _



 _ 

   

 

in which M11M11 q p0, 0,

(3.11) and the above estimate

k  K D From (3.10),

, we obtain

0,  u D0, 0 as n .

int theorem, there

ex

 

n n

C w_ w D_C u

On the basis of the Schauder fixed-po

ists a function

   



 

u z C D

 



such , and from Th

, ,

w z u z w z

 

 

at _w z u z

   

, _T w z u z_

   

, _

th

sy to see t

eorem

2.2, it is ea hat w z

 

,

 

1₀

 

,2

p

W D , and

   

, w z u z

 

  is a solution of Problem Q

u z 

for the Equa-tion (1.6) and the relaEqua-tion (1.8) with the condition 0,

1

 .

In addition, if G z u w



, ,



ReB w B u1 2



  in D, where 0  1,L_p_,2_B D_j, _k₀ , j1, 2, then the above solvability result still

similar method.

2) Secondly, we discuss the case:

hold by using the above





min  , 1. In this case, (3.5) has the solution tM10 provided that

M9 in (3.3) is small enough. Now we consider a closed

and convex subset B in the Banach space

   

,

C D C D i.e.

 

   



, ,



.

B w z u z C D C w D , C u D , M10 (3.16) Applying a method similar as before, we can verify that there exists a solution

   

, p₀,2

 

p₀,2

 

w z u z W D W D

 

 

) with the condition

1 1



 ,



 .

of Problem Q for (1.6 min

over, if





1 More



1 2

, , Re

G z u w  B w B u  in D, where

1   , Lp,2B Dj, k0 , j = 1, 2. Under the same

condition, we can derive the above solvability result by the similar method.

3) When G z u w



, ,



satisfies the condition (3.4), we can verify the uniqu ess of solutions in this ten heorem. In fact, if w z1

   

,u z1   , w z u z2

   

, 2  are two solu-

tions of Problem Q for the Equation (1.6), then

   

, 1

 

2

   

, 1 2

 

w z u z  w z w z u z u z

   

  

sa



tisfies the equation and boundary conditions



1 1



Re

z z

w  _Qw  A B w_





2 2 , ,

A B u z D



   

  _(3.17)

   

 

Re z w z  c u h z z1  , , (3.18)

 

 

 

0, .

j j

a w a j J



 _{ } _

 

 

in which

Im (3.19)

0 1

Q q  _{. Similarly to Theorem 2.2, we can}

derive the following estimates of the solution

   

w z u z,

 

  r cofo mplex Equation (3.17):

 

,

 

, 12 ,

C__w z D__C u z D__ __M k

    (3.20)

0,2 , 13 ,

p z z

L _w _w D__M k

  (3.21)

where



0



min ,1 2 p ,

   



0, 0, , , ,0



, 12,13



j j

M M q p k  K D j

are two non-negative constants, k2k C u0



,

over the estimate



.

D More-





 

, 1 2 0 13

C__w z D_  k M C u z__ ,D0

can be derived. Provided that the positive constant

 (3.22)

(6)

small enough such that



1 2 k M0 13



0, from (3.22) it

follows u z

 

u z1

 

u z2

 

0, i.e. u z1

 



 

D. Th e proof of the theorem.

bove theorem, the next result can

2

u z in

be de-

itions as in

for (1.1) has N

of Problem P

stants. is co

From rived.

rem 3.2, t

depends o 2) Whe

mpletes th the a

Theorem 3.3. Under th

he following statem en the index K > N

y conditions, and n 2K N 2 arb

n 0 ,

e same cond ents hold. , Problem P

the solution itrary real con

Theo-1) Wh solvabilit

K N

  Problem P for (1.1) is solvable,

if 2N K solvability conditions are satisfied, and the solution of Problem P depends on K2 arbitrary real co

is solvable under 2

nstants.

3) When K < 0, Problem P for (1.1)

2 1

N K conditions, and the solution of Problem P

depends on 1 arbitrary real constant.

Moreover, we can write down the solvability

condi-tio lem P for a

L

dary

tion (1.8). If the function h z

 

0,z , i.e.

1, ,

0,1, , , 0,

j m

N

N h m



 

 

ns of Prob ll other cases.

Proof. et the solution _w z u z

   

, _ of Problem Q for (1.6) be substituted into the boun condition (1.7) and the rela

0, 0, j

h j

 



, K



 _{1, ,}_ _{ }_K _{1, if}if 0_K_K_0,N,

and d_j0, j1, , N, then we have w z

 

u_z D

and the function w z

 

of Problem P

for (1.1). Hence the t r of above equalities is

just the number of solvability conditions as s this heorem. Also note that the real constants b0

in n

tated in in (1.8) and This shows that .1) includes the stated in the

theo-Classes of P

y atics Monograph American Mathematical Society, Providence, 1992. [5] G. C. Wen, D. C. Chen and Z. L. Xu, “Nonlinear

Com-plex Analysis and Its Applications,” Mathematics Mono-graph Series 12, Science Press, Beijing, 2008.

proximate Methods and Numerical Anal-

is otal num

in (1.9) are arbitrarily ution of Proble of arbitrary real consta

REF

.

just a solutio be

chosen. m P for (1

nts as t

b_j



j J



the general sol number

m.

[2] G. C

re

ERENCES

[1] I. N. Vekua, “Generalized Analytic Functions,” Pergamon, Oxford, 1962.

Wen and H. Begehr, “Boundary Value Problems for Elliptic Equations and Systems,” Longman Scientific and Technical Company, Harlow, 1990.

[3] A. V. Bitsadze, “Some artial Differential Equations,” Gordon and Breach, New York, 1988. [4] G. C. Wen, “Conformal Mappings and Boundar Value

Problems,” Translations of Mathem s 106,

[6] G. C. Wen, “Ap

ysis for Elliptic Complex Equations,” Gordon and Breach, Amsterdam, 1999.

G. C. Wen, “Linear and Quasilinear Complex Equ

[7] ations

of Hyperbolic and Mixed Type,” Taylor & Francis, Lon-don, 2002.doi:10.4324/9780203166581

G. C. Wen, “Recent Progress in Theory and Applica

[8] tions

of Modern Complex Analysis,” Science Press, Beijing, 2010.