УДК 519.63

16 қатарлары түрінде жазып аламыз. Мұндағы,

МРНТИ 27.29.19 УДК 519.63

МРНТИ 27.29.19

M.Н. Конуркулжаева

PhD докторант Казахского национального университета им. Аль-Фараби, Алматы, Казахстан МНОГОСТРУКТУРЫ В ПРОБЛЕМЕ ИХ ЗНАЧЕНИЙ СТЕЛОВОЙ ПРОБЛЕМЫ

Аңдатпа

Әр түрлі ақырлы өлшемді денелерді біріктірудегі шекаралық есептер көптеген инженерлік қосымшаларда кезедеседі, атап айтқанда, механикалық құрылымдар мен машиналардың серпінділігі мен беріктігін талдау және олардың сенімділігін болжау кезінде. Әр түрлі ақырлы өлшемді денелерді біріктірудің геометриясы күрделі болып табылады, бұл тиісті шекаралық есептерді тиімді шешуді қиындатады. Техникалық әдебиеттерде әртүрлі ақырлы өлшемді денелерді біріктірудің есептерінің сипаттаудың жеңілдетілген әдістерін табуға болады. Ең қарапайым әдістердің бірі ақырлы өлшемді модельді қамтиды, мұнда құрылымдар салмақты серпімді шектеулерге ие дискретті массалар жүйесі ретінде ұсынылады. Неғұрлым лайықты сипатталатын модельдер құрылымды серпімді инерциалды және диссипативті (пластиналар, қабықшалар), сондай-ақ үш өлшемді (қалың дискілер, қалың құйылған цилиндрлер және т.б.) біріктірілген серпімді денелер жүйесі ретінде қамтиды. Жіңішке стерженбдар, пластиналар және қабықшалар үшін асимптоталық теориялар зерттеушілермен ерте бастан назарын аударғанымен, әр түрлі ақырлы өлшемді денелерді біріктірудің асимптоталық талдау мәселесі салыстырмалы түрде жақында пайда болды. Бұл есептердің негізі Ciarlet және Destuynder (1979) атты ғалымдармен айтылған және әлі күнге дейін көптеген қызықты сұрақтарды тудырады. Ұсынылып отырған мақалада Стеклов есебінің меншікті мәні мен меншікті функцияларының асимптотикасы ені жұқа және өлшемдері бірге тең тіктөртбұрыштардың біріктірілуінде қарастырылады.

Түйін сөздер: Стекловтың спектральды есебі, меншікті мәндер, әр түрлі ақырлы өлшемді денелер, Соболев кеңістігі, Лебег кеңістігі, асимтотикалық түрі.

1. Statement of the problem

Defined a region ^, that consists of two rectangles: q^  ( 1, 0] ( 



, 0) _and Q(0, ] (l  d, 0) _which are thin and size is equal to one. Where



(0,1] the small parameter, Cartesian coordinates

(x, y)R2 _and

values l d,  are dimensionless due to the scaling of the long side _{(x, y) :y0,x ( 1, 0)} of the rectangle q^. We will consider the Steklov problem

( , ) 0, u x y^

  ( , )x y ^, ₍₁₎

( , 0) ( , 0),

yu x^



^{ }u x

  ( , )x y _,

( , ) 0,

vu x y^

  ( , )x y ^ \ ( T), where

2 2

x y

    

the delta operator,



^ the spectral parameter, {(x, y) :y0,x ( 1, )}l _, _v is the derivative along the outward normal, defined everywhere on the boundary ^ , except for the set P of corner points. Note that   ^{ y}

on . Problem (1.1) describes waves on the surface of heavy water in a reservoir with a shallow. Following [1], we enter the Sobolev space

1( )

H ^ a specific scalar product

, ( , ) ( , ) ,

u z  u z  u z 

 _



_

     

(1.2)

where  grad, (, )_

is the natural scalar product in the Lebesgue space L²(^).

The norm, generated by the scalar product (1.2), we denote by || || . _{ Let the} T^ operator in

1( )

H ^ , which is defined by the formula

, ( , ) ,

T u z^{  } _ u z^{ } _

   u z^, ^H¹(^). _(1.3)

The generalized formulation of problem (1.1) applies to the integral identity [2]

q^ε

ε Γ^ε

(u,z)_ 



(u z, ) ,_D zH1(),

(1.4) Which is according to the definitions (1.2) and (1.3), takes the form of the abstract equation

T u^{ } 



^{ }u _вH¹(^) _(1.5)

with a new spectral parameter

( ) .1

 





 

 ^ _(1.6)

The operator T^ is compact, positive and symmetric, hence self-adjoint, and its spectrum is located on a segment [0,



^1]. The discrete spectrum forms an infinitely small sequence of eigenvalues

1

0^ 1^ 2^ ... _n^ ... 0.



^ 











 



  

(1.7) A point



0 belongs to an essential spectrum and is an eigenvalue of infinite multiplicity with a proper subspace composed of those functions

1( ),

u^H ^ that turn into zero on D. The connection (1.6) of the spectral parameters rewrites the sequence (1.7) into a monotonically increasing unbounded sequence of Steklov eigenvalues (1.1)

0 1 2

0



^ 



^ 



^  ...



_n^   ... . _(1.8) The first eigenvalue



^1 0

is a prime number, and a constant eigenfunction corresponds to it. The point



0 is transffered to infinity, i.e. does not affect the spectrum of the boundary-value problem (1.1), which turns out to be entirely discrete.

The eigenvectors

1 0, 1,..., _n,... ( )

U U^{ } U^ H ^ of the operator T^ can be subordinated to orthogonality and normalization conditions

, 0

, , , {0},

m n m n

U U^{ }



m n N N

    

(1.9) where



^{m n,} 

Kronecker's symbol, a N is a natural sequence. Putting

( ) 1/2

n n n

u^ 

 

 ^ U^

taking into account the relations (1.2) and (1.4) , we obtain formulas

, 0

(u u_mⁿ, _nⁿ)



_{m n}, ,m nN .

(1.10) The aim of this work is to construct the asymptotics of the eigenvalues (1.8) for



 0.

Figure 1.

1. The formal asymptotic behavior in the low-frequency range Following [2]1, we take asymptotic ansatzes

2 1 1

...,u x( ) v x( ) v x( , y) ..., (x, y) q ,

  







  

 

^   _(2.1)

in which the number  and functions , 1

v v belong to the definition, and the dots denote the lower terms that are not essential for the formal analysis being undertaken. We substitute the expansion (2.1) into the problem (1.1),

more precisely into the differential equation constricted to a rectangle q^ , and to boundary conditions restricted to its long sides  _ {(x, y) : y0, x ( 1, 0)} _and  ^_{_} {(x, y) : y0, x ( 1, 0)}

(Fig. 1). Entering a fast variable

1y

 

 ^ and collecting the coefficients for the same powers of the small parameter , we arrive at the Neumann problem for an ordinary differential equation with parameter x ( 1, 0)_.

2 1 2 1 1

( , ) ( ), ( 1, 0), ( , 0) ( ), ( , 1) 0.

v x v x v x V x v x















         

(2.2) The solvability condition for problem (2.2) takes the form of equation

2 ( ) ( ), ( 1, 0),

xv x



v x x

    _(2.3)

which is complemented by the boundary conditions

( 1) 0

xv

  

(2.4) Proceeding from the Neumann boundary condition at the end of



^ {(x, y) :x 1,y (



, 0)} _the rectangle q^_.

To determine the boundary condition at a point x0 , we consider one more asymptotic ansatz on a large rectangle:

( , ) ( , ) 1( , ) ..., .

u x y^ V x y 



V x y  xQ _(2.5) Substituting it into the problem (1.1), constriction to Q, we find that V the solution of the homogeneous Neumann problem, i.e. V is constant. The next problem of Neumann

1 1

( ) 0, ( , ) , _v ( , ) 0, ( , ) \ ( ),

V x, y x y P V x y x y Q P

       

(2.6)

1( , 0) ( ) : , ( , ) (0, ) {0},

yV x G x



V x y _ l

     

(2.7)

Does not have a bounded solution in the case



V 0. Let us choose a solution with a logarithmic singularity at the origin , _i.e.

~

1 2 1

( , ) ln ( ),

V x y K r V x, y

   ^~1 ^~1

|V x, y( ) |cr,|V x, y( ) |c; (2.8) Here ( , )r



 the system of polar coordinates, r0 _and



(0,



/ 2) (Fig. 1). We note that the solution V¹ is determined up to a constant summand, and it is fixed so that

~

1(0, 0) 0;

V  pointwise estimates of the remainder

~

V1 and its derivatives are provided, for example, by general results of the theory of elliptic boundary value problems in domains with angular and canonical points [3], [4]. The coefficient K is computed using the method from [5].

Figure 2.

We substitute the functions V¹and 1 in the Green formula on the set Q_ {(x, y) :r



}

and pass to the limit with



 0._{We have}

/ 2

1 1

0 0

0

lim ( , 0) lim ( , ) | .

l

y r r

Vl V x dx V x y d K



  



  _ 

 





 



 

(2.9) In the framework of the method of matched asymptotic expansions ([6]14, [7]15) we integrate (2.1) (2.5) as external expansions and construct the following inner expansion in uniformly extended coordinates

1 1

( , )

 

(



^ x,



^ y) :

( , ) ( , ) 1( , ) ...

u x y^ w

  

 w

 

 _(2.10) The extension of the coordinates ( , )x y ( , )

 

and the formal transition to



0transforms the region ^ into the articulation  of the fourth quadrant

{( , ) 2: 0, 0}

B

 

R







 and half-strip ( , 0] ( 1; 0)

     (Fig. 2). Because the

1 1

yu^



^{ }u



^ _w _w



w ...

       

on ,

Both functions w _and w¹ satisfy the homogeneous Neumann problem in the domain _. Realizing the splicing of the expansions (2.10) and (1.1), we see that

1 2

( , ) ~ V, ( , ) ~ K ln

w  w   

 _with   ²²  ^{, ( , )}  ^B _(2.11) As a result, we find that w( , )

 

V everywhere in  _and

1( , ) KW( , ),

w

 



 

_(2.12)

where W is harmonic in a  function satisfying the Neumann condition on  and admits the following representations in an angle and a half-strip:

2 1

( , ) ln ( ), , ( , ) ,

( , ) (e ), , ( , ) .

W O B

W b O ^

      



     

     

      _(2.13)

A function W can be constructed using a conformal transformation, in particular, b an absolute constant, which, like the explicit form of the function itself, will not be needed later. The coupling of the coefficients in the representations (2.13) of the function W is determined by means of an analogous (2.9) calculation, that is, according to [5], with the application of the Green formula in the region ^R

(deeply toned in Fig. 2) and the limiting transition R _.

We draw attention to the fact that lnlnrln ,



and in order to perform splicing with exactness ( 2) O



_{, the}

second term on the right-hand side of (2.5) must be replaced by the sum

1 2

( , ) | ln | .

V x y K 

 

Next we see that the additional constant in expression (2.14) does not affect on subsequent calculations.

In the immediate vicinity to the point ,to the external expansion from (2.1) can be given a form ( ) (0) _y (0) ...

u x^ v  



v 

(2.15) Comparing the representation (2.15) with the first formula (2.11) and the second formula (2.13), with the equality (2.9) taken into account, we conclude that V v(0),K  _xv(0)

and

(0) (0).

xv



lv

 

Let us formulate the resulting spectral problem (2.3), (2.4), (2.16), the eigenvalues



⁰ 0

and



_j 0, jN,

which are the roots of the transcendental equation

0, l



tg





(2.17)

A (non-normalized) eigenfunctions have the form

0( ) 1, _j( ) _jsin( _j ) cos( _j ), . v x  v x tg   x   x jN

(2.18) References

1 NazarovS.А.Concentration of trapped modes in problems of the linearized theory of water waves //Mat.

Sb., 2008, Volume 199, Number 12, Pages 53–78

2 Ladyzhenskaya O.A. Boundary value problems of mathematical physics. Moscow: Nauka, 1973.

3 Maz'ya V.G., Plamenevsky B.A. Estimates in L^p

and in Holder classes and the Miranda maxima principle for solutions of elliptic boundary value problems in domains with singular points on the boundary // Math. Nachr.

1977. Bd. 77. S. 25-82.

4 NazarovS.A.,PlamenevskyB.A.Ellipticproblemsindomainswithpiecewise smoothboundaries.Berlin,NewYork:WalterdeGruyter.1994.

5 Maz'ya V.G., Plamenevsky B.A. On coefficients in the asymptotics of solutions of elliptic boundary value problems in a domain with conical points / Math. Nachr. 1977. Bd. 76. S. 29-60.

6 Van Dijk M.D. Methods of perturbations in fluid mechanics. Moscow: The World, 1967.

7 Ilin A.M. Harmonization of asymptotic expansions of solutions of boundary value problems. M .: Nauka, 1989.

ГРНТИ 28.17.31