Уравнения математической физики (краевые задачи в пространствах Соболева). Салехов Л.Г - 15 стр.

dALEE,
         Zt                                       Zt                           1            1
              kum(s)kH      2
                                1
                                0 ()
                                        ds = (f (t)jum(s))ds ;
                                                                               2
                                                                                 k um (t)k + kum (0)k
                                                                                          2
                                                                                            2
                                                                                                      2

          0                                       0

ILI
 Zt                                      Zt                                              Zt
        kum(s)k2H01()ds 6 (f (s)jum(s))ds + 21 kum(0)k2 6                                      kf k  kumkds+
    0                                    0                                                  0
                              0Z t     Z t      11=2
              + 1 kum (0)k2 6 @ kf k2ds kum k2dsA + 1 kum (0)k2:
                2                                                                       2
                                                  0                 0

nO, W SILU NERAWENSTWA fRIDRIHSA, IMEEM: kum k2 6 k2 kumk2H (). tOGDA                               1
                                                                                                     0


Zt                        0Z t        Zt                11=2
        kum(s)k2H ()ds 6 @ kf (s)k2ds k2kum(s)k2H ()dsA + 1 kum(0)k2:
                    1
                    0
                                                                                    1
                                                                                    0                2
0                                            0                      0

  tEPERX ISPOLXZUEM SLEDU@]IJ FAKT: ESLI a b c | TRI POLOVITELX-
NYH ^ISLA, TAKIH, ^TO a2 6 ab + c2=2, TO a2 6 b2 + c2. iMEEM:
                    Zt                                              Zt
                            kum(s)kH                     ds 6 k 2        kf (s)k ds + kum(0)k :
                                                                                        N
                                             2                                 2                 2
                                                 1
                                                 0 ()
                        0                                           0

|TO NERAWENSTWO POKAZYWAET, ^TO DLQ KAVDOGO m 2 um ESTX \LEMENT
IZ L2(I  H01()) I ^TO
                  kumk2L (IH ()) 6 k2kf kL (I) + ku0mk2:
                                             2        1
                                                      0
                                                                           2


    c) sHODIMOSTX PRIBLIVENNYH REENIJ.
    wYBEREM u0m TAK, ^TOBY POSLEDOWATELXNOSTX (u0m)m2N SHODILASX K
u0 (x) PO TOPOLOGII H01(). tOGDA POSLEDOWATELXNOSTX (ku0m k2)m2N SHO-
DITSQ.
    nAPOMNIM, ^TO WSQKOE GILXBERTOWO PROSTRANSTWO MOVET BYTX IZO-
METRI^ESKI OTOVDESTWLENO SO SWOIM SOPRQVENNYM PROSTRANSTWOM (W
SILU TEOREMY rISSA). pO\TOMU NA \TOM GILXBERTOWOM PROSTRANSTWE
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