О ВАРИАЦИОННОМ РАВЕНСТВЕ ДЛЯ ФУНКЦИОНАЛА ОБЩЕГО ВИДА

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<ul><li><p> 517.95 22.162</p><p> 1</p><p>.. </p><p> , -, , - . - - . , .</p><p> : , -, , , .</p><p>1. </p><p> 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, ).</p><p> ~u = (u1, ..., um) Liploc(), - ~u D~u = (u1, ...,um).</p><p>I(~u) =</p><p>G(x, ~u(x), D~u(x))dx (1)</p><p> - ~u = (u1, ..., um) Liploc(), , (1).</p><p> , G(x, z, ) , (., , [1, . 6, . I]) </p><p>Qi[~u] nh=1</p><p>d</p><p>dxh</p><p>(Gih(x, ~u,D~u)</p><p>)Gzi(x, ~u,D~u) = 0, i = 1, ...,m. (2)</p><p> 1 = (11 , ..., 1n), ..., </p><p>m = (m1 , ..., mn ).</p><p>18</p><p>..,2012</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17)</p></li><li><p> - (., -, [5; 8]). , , -. , [2; 3]. [2] .</p><p> G(z, , w, ) G, - (z, ) (w, ). , , , (1) - ~u,~v, ~u , , G(~u,D~u,~v,D~v). - (2) G(z, , w, ). - p- [2].</p><p> 1. , m = n </p><p>G(x, z1, ..., zm, 1, ..., n) =i,j,h,k</p><p>ahkij (x)hi </p><p>kj +</p><p>+mi=1</p><p>F i(x)zi,</p><p> ahkij = akhji = a</p><p>ikhj </p><p>1i,h</p><p>hi hi </p><p>i,j,h,k</p><p>ahkij (x)hi </p><p>kj 2</p><p>i,h</p><p>hi hi , </p><p>hi = </p><p>ih,</p><p> (2) . :</p><p>j,h,k</p><p>xh</p><p>(ahkij (x)</p><p>uj</p><p>xk</p><p>)= F i(x), i = 1, ..., n. (3)</p><p>, (3) , , [6] [9], [9] .</p><p> 2. </p><p>G(x, z1, ..., zm, 11 , ..., mn ) =</p><p>i,h</p><p>(hi )2,</p><p> (2) u1 = 0, ...,um = 0</p><p> Rn Rm.</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 19</p></li><li><p> 3. , m = 1 </p><p>G(x, z, ) = G(x, z, 1, ..., n),</p><p>nh=1</p><p>d</p><p>dxh(Gh(x, u,u)) = Gz(x, u,u).</p><p> , G(x, z, ) = ||2, . G(x, z, ) ==</p><p>1 + ||2, . , </p><p>G(x, z, ) =</p><p>1 ||2, , -- .</p><p> (2). 1. - -- ~ = (1, ..., m) L2() ~ = (1, ..., m) C( \ ). - ~u == (u1, ..., um) Lip() (2), </p><p>1) ~u C( ( \ )) ~u|\ = ~|\;</p><p>2) - ~v = (v1, ..., vm) Liploc( ( \))L2(), , ~v = 0 \ , </p><p>nh=1</p><p>Gih(x, ~u(x), D~u(x))vixhdx</p><p>vii ds+</p><p>Gzi(x, ~u(x), D~u(x))vi(x)dx = 0</p><p> i = 1, ...,m.</p><p>. -, G(x, z, ) - u C2() C1() - , , u (2), </p><p>~u|\ = ~|\ ,nh=1</p><p>Gih(x, ~u,D~u)h| = i| , i = 1, ...,m,</p><p> = (1, ..., n) .</p><p>, = , = .</p><p> (z, ), (w, ) Rm Rmn </p><p>G(z, , w, ) = G(x,w, )G(x, z, )nh=1</p><p>mi=1</p><p>Gih(x, z, )(ih ih)</p><p>20 .. . </p></li><li><p>mi=1</p><p>Gzi(x, z, )(wi zi).</p><p>, G(x, z, ) (z, ), (w, ) (z, ), G(x, z, ) z. , G z , </p><p>G(z, , w, ) &gt; 0, 6= z 6= w. 4. m = 1. </p><p>G(x, z, ) =n</p><p>i,j=1</p><p>aij(x)ij,</p><p> ||aij(x)|| . </p><p>G(z, , w, ) =n</p><p>i,j=1</p><p>aij(x)(i i)(j j).</p><p> , G(x, z, ) = ||2, G(z, , w, ) = | |2. - .</p><p> 1. - ~u Liploc( ( \ )) (2) - ~ L2() - ~v Liploc( (\))L2(), </p><p>~u|\ = ~v|\. </p><p>G(~u,D~u,~v,D~v)dx = I(~v) I(~u)</p><p>mi=1</p><p>i(vi ui) ds.</p><p>. z = ~u, w = ~v, = D~u, = D~v G(z, w, , ), , </p><p>G(~u,D~u,~v,D~v)dx =</p><p>G(x,~v,D~v)dx</p><p>G(x, ~u,D~u)dx</p><p>nh=1</p><p>mi=1</p><p>Gih(x, ~u,D~u)(vixh uixh)dx</p><p>mi=1</p><p>Gzi(x, ~u,D~u)(vi ui) =</p><p>=</p><p>G(x,~v,D~v)dx</p><p>G(x, ~u,D~u)dx</p><p>mi=1</p><p>i(vi ui) ds = I(~v)</p><p>I(~u)</p><p>mi=1</p><p>i(vi ui) ds.</p><p> .</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 21</p></li><li><p>. , </p><p>I1(~v) = I(~v)</p><p>mi=1</p><p>ivi ds,</p><p> 1, , , ~u I1 . , {~vm}, , I1(~vm) I1(~u), </p><p>G(~u, ~vm, D~u,D ~vm)dx 0.</p><p> (vm, D~vm) (u,D~u) ., m = 1 G(x, z, ) = ||2, </p><p>|uvm|2dx 0,</p><p> vm u W 1,2(). , [3] - vm.</p><p> .</p><p> 1. 1, -~v (2). </p><p>G(~u,~v,D~u,D~v) + G(~v, ~u,D~v,D~u) = 0.</p><p> , G(x, z, ) z , ~u = ~v + const. , 6= , const = 0.</p><p>. </p><p>G(~u,~v,D~u,D~v)dx = I1(~v) I1(~u),</p><p>G(~v, ~u,D~v,D~u)dx = I1(~u) I1(~v).</p><p> , </p><p>(G(~u,~v,D~u,D~v) + G(~v, ~u,D~v,D~u))dx = 0.</p><p> . G z, ~u = ~v. G , D~u = D~v. ~u ~v , , ~u ~v const.</p><p>22 .. . </p></li><li><p>. m = n G(x, z, w, , ) = G(x, , ). , - </p><p>G(, ) 0n</p><p>i,j=1</p><p>(ij ij)2</p><p> 0 &gt; 0 , , ij = ji , </p><p>ij = </p><p>ji , G(, ) </p><p> . - (., , [6]).</p><p>. [4] G(, )+G(, ) </p><p>ni=1</p><p>xi</p><p>(|u|p2uxi</p><p>)= 0,</p><p> m = 1 G(x, z, ) = ||p. .</p><p> 1 .</p><p>2. </p><p> , .</p><p> Rn. , x Rn (x, ), :</p><p>1) (x, ) 0;2) (x, ) = (x, ) &gt; 0;3) </p><p>(x) = { Rn : (x, ) &lt; 1} x .</p><p>H(x, ) = sup 6=0</p><p>, (x, )</p><p>. (4)</p><p> (. [7, 15])</p><p>(x, ) = sup 6=0</p><p>, H(x, )</p><p>. (5)</p><p> , H(x, ) 1)3), (x, ).</p><p> , 2(x, ) H2(x, ) Rn. </p><p>A = A(x, ) = (x, ), 6= 0,</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 23</p></li><li><p>B = B(x, ) = H(x, ), 6= 0., -</p><p>A(x, ), = (x, ), B(x, ), = H(x, ) (6) , Rn.</p><p>. , (x) . (4) (5) = () = () , </p><p>(x, ()) = 1, H(x, ()) = 1.</p><p> A(x, ) B(x, ) .</p><p> 1. </p><p>A(x, ) = (), B(x, ) = ().</p><p>. Rn (2) </p><p>(x, + t) (x, ) + t, () , () =</p><p>= t, ().,</p><p>A(x, ), , (). . - .</p><p> 2. , Rn </p><p>A(x, ), (x, ).</p><p> , = &gt; 0.</p><p>. 1, </p><p>A(x, ), H(x,A(x, ))(x, ) =</p><p>= H (x, ()) (x, ) = (x, ).</p><p> , </p><p>H(x, ()) = 1 =(), (x, )</p><p>.</p><p> , () </p><p>(x, ) =(), H(x, ())</p><p>24 .. . </p></li><li><p>H(x, ()) =(), (x, )</p><p>.</p><p> 2 </p><p>(x, )=</p><p>(x, ).</p><p> .</p><p> , Rn </p><p>(, ) = 2(x, ) + 2(x, ) 2(x, )A(x, ), .</p><p>, (, ) 0 (, ) = 0 , = . , 2 </p><p>(, ) 2(x, ) + 2(x, ) 2(x, )(x, ) =</p><p>= ((x, ) (x, ))2 0. , , , -,</p><p>(x, ) = (x, ),</p><p> -, A(x, ), = (x, ). 2 , </p><p>(x, )=</p><p>(x, ),</p><p> = .</p><p> 5. </p><p>(x, ) =</p><p>(n</p><p>i,j=1</p><p>gij(x)ij</p><p>)1/2, = (1, ..., n),</p><p> (gij(x)) . </p><p>H(x, ) =</p><p>(n</p><p>i,j=1</p><p>gij(x)ij</p><p>)1/2, = (1, ..., n),</p><p> gij(x) (gij(x)). </p><p>(, ) =n</p><p>i,j=1</p><p>gij(x)(i i)(j j).</p><p> 3. 6= 0 </p><p>A(x,B(x, )) =</p><p>H(x, ).</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 25</p></li><li><p>. 1 , B(x, ) = (). </p><p>(x, ()) = 1 H(x, ) = (), . 1 1 , </p><p>(x,B(x, )) = H(x, )</p><p>, B(x, ),</p><p> . .</p><p>3. </p><p> .</p><p> a(x) , . </p><p>(f) =</p><p>1 + 2(x,f) a(x) dx.</p><p> , , (.,, [1, . 6 ])</p><p>div</p><p>((x,f)A(x,f)a(x)</p><p>1 + 2(x,f)</p><p>)= 0. (7)</p><p> , (x, ) = ||, a(x) 1, (f) - f Rn+1, (7) .</p><p> f C1() </p><p>f =(f,1)</p><p>1 + 2(x,f).</p><p> = (, t), = (, t) Rn+1 </p><p>(, ) = (, ) + (t t)2. 2 , (, ) 0 (, ) = 0 , = .</p><p> 2. , f C2() C1()C() (7) g C2()C1()C(), f = g \ </p><p>(x,f)A(x,f), ~n1 + 2(x,f)</p><p>=(x,g)A(x,g), ~n</p><p>1 + 2(x,g)= 0</p><p> . </p><p>(f , g)</p><p>1 + 2(x,g) a(x) dx = 2((g) (f)). (8)</p><p>26 .. . </p></li><li><p>. 1. , </p><p>(f , g)</p><p>1 + 2(x,g) = 2</p><p>1 + 2(x,g)</p><p>2(x,f)A(x,f),g+ 11 + 2(x,f)</p><p>=</p><p>= 2</p><p>(1 + 2(x,g)</p><p>1 + 2(x,f)</p><p>(x,f)A(x,f),g f1 + 2(x,f)</p><p>)= 2G(f,g).</p><p> G </p><p>G(x, ) =</p><p>1 + 2(x, ).</p><p> a(x), - 1. .</p><p>. , (x, ) = ||, a(x) 1. (7) . , (f , g) f g. 2 </p><p>sin2</p><p>2</p><p>1 + |g|2dx = 1</p><p>2((g) (f)),</p><p> f g (x, f(x)) (x, g(x)) .</p><p>4. p-</p><p> u C1() C0,1() p &gt; 1 </p><p>Ip(u) =</p><p>p(x,u) dx+</p><p>F (x)u(x)dx,</p><p> F (x) , . </p><p>div(p1(x,u)A(x,u)</p><p>)= F (x), (9)</p><p> Ip(u). , Rn Rn,</p><p>p(, ) = (p 1)p(x, ) + p(x, ) pp1(x, )A(x, ), .</p><p>, 2(, ) = (, ).</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 27</p></li><li><p> 4. , Rn p(, ) 0. , = .</p><p>. 2 , </p><p>p(, ) (p 1)p(x, ) + (x, ) pp1(x, )(x, ) (p 1)p(x, ) + p(x, ) (p 1)p(x, ) p(x, ) = 0.</p><p> , , 2 , = &gt; 0. </p><p>0 = p(, ) = (p 1 + p p)p(x, ). p p &gt; 1 , </p><p>p 1 + p p = 0 = 1. , = .</p><p> 3. u C2() C0,1() (9) v C2() C0,1() , u = v . </p><p>p(u,v) = Ip(v) Ip(u). (10)</p><p>. 1 </p><p>p(, ) = (p 1)p(x, ) + p(x, ) pp1(x, )A(x, ), == p(x, ) p(x, ) pp1(x, )A(x, ), = G(, ),</p><p> . G G(x, ) = p(x, ). .</p><p>, - (9).</p><p> 4. u, v C2() C() (9). u(x) v(x) + sup</p><p>(u v).</p><p>. x0 , u(x0) &gt; v(x0) + sup</p><p>(u v). k = sup</p><p>(uv). &gt; 0 , u(x0) &gt; v(x0)+k+ </p><p> D {x : u(x) &gt; v(x) + k + }, x0. D , u, v + k + (9) D u = v + k + D. 3 </p><p>D</p><p>p(u,v)dx =D</p><p>p(x,u)dxD</p><p>p(x,v)dx,</p><p>D</p><p>p(v,u)dx =D</p><p>p(x,v)dxD</p><p>p(x,u)dx.</p><p> , </p><p>28 .. . </p></li><li><p>D</p><p>p(u,v)dx+D</p><p>p(v,u)dx = 0.</p><p> 4, u = v D. , u v+k+, D. .</p><p>1 , 11-01-97021-_ _ .</p><p>1. , . . : / . . ,. . , . . . . : , 1986. 760 c.</p><p>2. , . . / . . // . . XXVI. . : - , 2005. C. 201208.</p><p>3. , . . , - / . . // . : - , 2007. . 2. C. 136142.</p><p>4. , . . / . . . :- , 2008. 424 c.</p><p>5. , . . / . . . . :, 1970. 512 c.</p><p>6. , . . / . . , . . , . . . . : - , 1990. 311 c.</p><p>7. , . / . . . : , 1973. 471 c.8. , . . -</p><p> / . . . . : , 1988. 336 c.9. , . / . . . : , 1974. </p><p>149 c.</p><p>THE VARIATIONAL EQUALITY FOR THE FUNCTIONALOF THE GENERAL FORM</p><p>A.A. Klyachin</p><p>We defined a characteristic that serves as a measure of the difference of two vectorsassociated with a convex function. In terms of this characteristic we establish the variationalequality for the extremals of the functional of general form. We consider the results for theminimal surface equation in Finsler space and give them a geometric interpretation. Corollaryof results is the convergence on average sequence that minimizes this functional.</p><p>Key words: variational problem, minimization of a convex functional, the minimalsurface equation, mixed boundary value problem, Finsler metric.</p><p>ISSN 2222-8896. . . . -. . 1, . . 2012. 2 (17) 29</p></li></ul>

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