О ВАРИАЦИОННОМ РАВЕНСТВЕ ДЛЯ ФУНКЦИОНАЛА ОБЩЕГО ВИДА

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  • 517.95 22.162

    1

    ..

    , -, , - . - - . , .

    : , -, , , .

    1.

    Rn G(x, z, ) , x , z = (z1, ..., zm) Rm = (1, ..., m) Rn. , G(x, z, ) C( Rm Rm), z1, ..., zm, 1, ..., m . , G z, G(x, ).

    ~u = (u1, ..., um) Liploc(), - ~u D~u = (u1, ...,um).

    I(~u) =

    G(x, ~u(x), D~u(x))dx (1)

    - ~u = (u1, ..., um) Liploc(), , (1).

    , G(x, z, ) , (., , [1, . 6, . I])

    Qi[~u] nh=1

    d

    dxh

    (Gih(x, ~u,D~u)

    )Gzi(x, ~u,D~u) = 0, i = 1, ...,m. (2)

    1 = (11 , ..., 1n), ...,

    m = (m1 , ..., mn ).

    18

    ..,2012

    ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17)

  • - (., -, [5; 8]). , , -. , [2; 3]. [2] .

    G(z, , w, ) G, - (z, ) (w, ). , , , (1) - ~u,~v, ~u , , G(~u,D~u,~v,D~v). - (2) G(z, , w, ). - p- [2].

    1. , m = n

    G(x, z1, ..., zm, 1, ..., n) =i,j,h,k

    ahkij (x)hi

    kj +

    +mi=1

    F i(x)zi,

    ahkij = akhji = a

    ikhj

    1i,h

    hi hi

    i,j,h,k

    ahkij (x)hi

    kj 2

    i,h

    hi hi ,

    hi =

    ih,

    (2) . :

    j,h,k

    xh

    (ahkij (x)

    uj

    xk

    )= F i(x), i = 1, ..., n. (3)

    , (3) , , [6] [9], [9] .

    2.

    G(x, z1, ..., zm, 11 , ..., mn ) =

    i,h

    (hi )2,

    (2) u1 = 0, ...,um = 0

    Rn Rm.

    ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 19

  • 3. , m = 1

    G(x, z, ) = G(x, z, 1, ..., n),

    nh=1

    d

    dxh(Gh(x, u,u)) = Gz(x, u,u).

    , G(x, z, ) = ||2, . G(x, z, ) ==

    1 + ||2, . ,

    G(x, z, ) =

    1 ||2, , -- .

    (2). 1. - -- ~ = (1, ..., m) L2() ~ = (1, ..., m) C( \ ). - ~u == (u1, ..., um) Lip() (2),

    1) ~u C( ( \ )) ~u|\ = ~|\;

    2) - ~v = (v1, ..., vm) Liploc( ( \))L2(), , ~v = 0 \ ,

    nh=1

    Gih(x, ~u(x), D~u(x))vixhdx

    vii ds+

    Gzi(x, ~u(x), D~u(x))vi(x)dx = 0

    i = 1, ...,m.

    . -, G(x, z, ) - u C2() C1() - , , u (2),

    ~u|\ = ~|\ ,nh=1

    Gih(x, ~u,D~u)h| = i| , i = 1, ...,m,

    = (1, ..., n) .

    , = , = .

    (z, ), (w, ) Rm Rmn

    G(z, , w, ) = G(x,w, )G(x, z, )nh=1

    mi=1

    Gih(x, z, )(ih ih)

    20 .. .

  • mi=1

    Gzi(x, z, )(wi zi).

    , G(x, z, ) (z, ), (w, ) (z, ), G(x, z, ) z. , G z ,

    G(z, , w, ) > 0, 6= z 6= w. 4. m = 1.

    G(x, z, ) =n

    i,j=1

    aij(x)ij,

    ||aij(x)|| .

    G(z, , w, ) =n

    i,j=1

    aij(x)(i i)(j j).

    , G(x, z, ) = ||2, G(z, , w, ) = | |2. - .

    1. - ~u Liploc( ( \ )) (2) - ~ L2() - ~v Liploc( (\))L2(),

    ~u|\ = ~v|\.

    G(~u,D~u,~v,D~v)dx = I(~v) I(~u)

    mi=1

    i(vi ui) ds.

    . z = ~u, w = ~v, = D~u, = D~v G(z, w, , ), ,

    G(~u,D~u,~v,D~v)dx =

    G(x,~v,D~v)dx

    G(x, ~u,D~u)dx

    nh=1

    mi=1

    Gih(x, ~u,D~u)(vixh uixh)dx

    mi=1

    Gzi(x, ~u,D~u)(vi ui) =

    =

    G(x,~v,D~v)dx

    G(x, ~u,D~u)dx

    mi=1

    i(vi ui) ds = I(~v)

    I(~u)

    mi=1

    i(vi ui) ds.

    .

    ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 21

  • . ,

    I1(~v) = I(~v)

    mi=1

    ivi ds,

    1, , , ~u I1 . , {~vm}, , I1(~vm) I1(~u),

    G(~u, ~vm, D~u,D ~vm)dx 0.

    (vm, D~vm) (u,D~u) ., m = 1 G(x, z, ) = ||2,

    |uvm|2dx 0,

    vm u W 1,2(). , [3] - vm.

    .

    1. 1, -~v (2).

    G(~u,~v,D~u,D~v) + G(~v, ~u,D~v,D~u) = 0.

    , G(x, z, ) z , ~u = ~v + const. , 6= , const = 0.

    .

    G(~u,~v,D~u,D~v)dx = I1(~v) I1(~u),

    G(~v, ~u,D~v,D~u)dx = I1(~u) I1(~v).

    ,

    (G(~u,~v,D~u,D~v) + G(~v, ~u,D~v,D~u))dx = 0.

    . G z, ~u = ~v. G , D~u = D~v. ~u ~v , , ~u ~v const.

    22 .. .

  • . m = n G(x, z, w, , ) = G(x, , ). , -

    G(, ) 0n

    i,j=1

    (ij ij)2

    0 > 0 , , ij = ji ,

    ij =

    ji , G(, )

    . - (., , [6]).

    . [4] G(, )+G(, )

    ni=1

    xi

    (|u|p2uxi

    )= 0,

    m = 1 G(x, z, ) = ||p. .

    1 .

    2.

    , .

    Rn. , x Rn (x, ), :

    1) (x, ) 0;2) (x, ) = (x, ) > 0;3)

    (x) = { Rn : (x, ) < 1} x .

    H(x, ) = sup 6=0

    , (x, )

    . (4)

    (. [7, 15])

    (x, ) = sup 6=0

    , H(x, )

    . (5)

    , H(x, ) 1)3), (x, ).

    , 2(x, ) H2(x, ) Rn.

    A = A(x, ) = (x, ), 6= 0,

    ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 23

  • B = B(x, ) = H(x, ), 6= 0., -

    A(x, ), = (x, ), B(x, ), = H(x, ) (6) , Rn.

    . , (x) . (4) (5) = () = () ,

    (x, ()) = 1, H(x, ()) = 1.

    A(x, ) B(x, ) .

    1.

    A(x, ) = (), B(x, ) = ().

    . Rn (2)

    (x, + t) (x, ) + t, () , () =

    = t, ().,

    A(x, ), , (). . - .

    2. , Rn

    A(x, ), (x, ).

    , = > 0.

    . 1,

    A(x, ), H(x,A(x, ))(x, ) =

    = H (x, ()) (x, ) = (x, ).

    ,

    H(x, ()) = 1 =(), (x, )

    .

    , ()

    (x, ) =(), H(x, ())

    24 .. .

  • H(x, ()) =(), (x, )

    .

    2

    (x, )=

    (x, ).

    .

    , Rn

    (, ) = 2(x, ) + 2(x, ) 2(x, )A(x, ), .

    , (, ) 0 (, ) = 0 , = . , 2

    (, ) 2(x, ) + 2(x, ) 2(x, )(x, ) =

    = ((x, ) (x, ))2 0. , , , -,

    (x, ) = (x, ),

    -, A(x, ), = (x, ). 2 ,

    (x, )=

    (x, ),

    = .

    5.

    (x, ) =

    (n

    i,j=1

    gij(x)ij

    )1/2, = (1, ..., n),

    (gij(x)) .

    H(x, ) =

    (n

    i,j=1

    gij(x)ij

    )1/2, = (1, ..., n),

    gij(x) (gij(x)).

    (, ) =n

    i,j=1

    gij(x)(i i)(j j).

    3. 6= 0

    A(x,B(x, )) =

    H(x, ).

    ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 25

  • . 1 , B(x, ) = ().

    (x, ()) = 1 H(x, ) = (), . 1 1 ,

    (x,B(x, )) = H(x, )

    , B(x, ),

    . .

    3.

    .

    a(x) , .

    (f) =

    1 + 2(x,f) a(x) dx.

    , , (.,, [1, . 6 ])

    div

    ((x,f)A(x,f)a(x)

    1 + 2(x,f)

    )= 0. (7)

    , (x, ) = ||, a(x) 1, (f) - f Rn+1, (7) .

    f C1()

    f =(f,1)

    1 + 2(x,f).

    = (, t), = (, t) Rn+1

    (, ) = (, ) + (t t)2. 2 , (, ) 0 (, ) = 0 , = .

    2. , f C2() C1()C() (7) g C2()C1()C(), f = g \

    (x,f)A(x,f), ~n1 + 2(x,f)

    =(x,g)A(x,g), ~n

    1 + 2(x,g)= 0

    .

    (f , g)

    1 + 2(x,g) a(x) dx = 2((g) (f)). (8)

    26 .. .

  • . 1. ,

    (f , g)

    1 + 2(x,g) = 2

    1 + 2(x,g)

    2(x,f)A(x,f),g+ 11 + 2(x,f)

    =

    = 2

    (1 + 2(x,g)

    1 + 2(x,f)

    (x,f)A(x,f),g f1 + 2(x,f)

    )= 2G(f,g).

    G

    G(x, ) =

    1 + 2(x, ).

    a(x), - 1. .

    . , (x, ) = ||, a(x) 1. (7) . , (f , g) f g. 2

    sin2

    2

    1 + |g|2dx = 1

    2((g) (f)),

    f g (x, f(x)) (x, g(x)) .

    4. p-

    u C1() C0,1() p > 1

    Ip(u) =

    p(x,u) dx+

    F (x)u(x)dx,

    F (x) , .

    div(p1(x,u)A(x,u)

    )= F (x), (9)

    Ip(u). , Rn Rn,

    p(, ) = (p 1)p(x, ) + p(x, ) pp1(x, )A(x, ), .

    , 2(, ) = (, ).

    ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 27

  • 4. , Rn p(, ) 0. , = .

    . 2 ,

    p(, ) (p 1)p(x, ) + (x, ) pp1(x, )(x, ) (p 1)p(x, ) + p(x, ) (p 1)p(x, ) p(x, ) = 0.

    , , 2 , = > 0.

    0 = p(, ) = (p 1 + p p)p(x, ). p p > 1 ,

    p 1 + p p = 0 = 1. , = .

    3. u C2() C0,1() (9) v C2() C0,1() , u = v .

    p(u,v) = Ip(v) Ip(u). (10)

    . 1

    p(, ) = (p 1)p(x, ) + p(x, ) pp1(x, )A(x, ), == p(x, ) p(x, ) pp1(x, )A(x, ), = G(, ),

    . G G(x, ) = p(x, ). .

    , - (9).

    4. u, v C2() C() (9). u(x) v(x) + sup

    (u v).

    . x0 , u(x0) > v(x0) + sup

    (u v). k = sup

    (uv). > 0 , u(x0) > v(x0)+k+

    D {x : u(x) > v(x) + k + }, x0. D , u, v + k + (9) D u = v + k + D. 3

    D

    p(u,v)dx =D

    p(x,u)dxD

    p(x,v)dx,

    D

    p(v,u)dx =D

    p(x,v)dxD

    p(x,u)dx.

    ,

    28 .. .

  • D

    p(u,v)dx+D

    p(v,u)dx = 0.

    4, u = v D. , u v+k+, D. .

    1 , 11-01-97021-_ _ .

    1. , . . : / . . ,. . , . . . . : , 1986. 760 c.

    2. , . . / . . // . . XXVI. . : - , 2005. C. 201208.

    3. , . . , - / . . // . : - , 2007. . 2. C. 136142.

    4. , . . / . . . :- , 2008. 424 c.

    5. , . . / . . . . :, 1970. 512 c.

    6. , . . / . . , . . , . . . . : - , 1990. 311 c.

    7. , . / . . . : , 1973. 471 c.8. , . . -

    / . . . . : , 1988. 336 c.9. , . / . . . : , 1974.

    149 c.

    THE VARIATIONAL EQUALITY FOR THE FUNCTIONALOF THE GENERAL FORM

    A.A. Klyachin

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