Уравнения математической физики (краевые задачи в пространствах Соболева). Салехов Л.Г - 18 стр.

tAK KAK EDINI^NYJ AR IZ L2(I  H01()) QWLQETSQ SEKWENCIALXNO KOM-
PAKTNYM PO OSLABLENNOJ TOPOLOGII, TO POSLEDOWATELXNOSTX (um)m2N
SHODITSQ K u(t x) PO OSLABLENNOJ TOPOLOGII L2(I  H01()), I NET NEOB-
HODIMOSTI IZ NEE IZWLEKATX PODPOSLEDOWATELXNOSTX.
  f ) sHODIMOSTX (um )m2N K u PO NORMIROWANNOJ TOPOLOGII
PROSTRANSTWA L2(I H01()).
  wYPIEM SNA^ALA SOOTNOENIQ \NERGII, USTANOWLENNYE RANEE, DLQ
um I u:
         (u0m(t)jum(t)) + kum (t)k2H () = (f (t)jum(t)) t 2 T
                                                    1
                                                    0


               (u0 (t)ju(t)) + ku(t)k2H01() = (f (t)ju(t)) t 2 T:
tAK KAK (um)m2N RSHODITSQ K u(t x) PO OSLABLENNOJ     R     TOPOLOGII
L (I H0 ()), TO 0 (f (t)jum(t))dt SHODITSQ K 0 (f (t)ju(t))dt. w SAMOM
 2      1             T                                  T
DELE, SU]ESTWUET g 2 L2(I H01()) TAKOE, ^TO ;g = f (t) NO TOGDA
                    ZT              ZT                      ZT
                         (f ju)dt = ; (gju)dt = (gju)H01()dt:
                    0               0                       0

   a TOGDA IMEEM:
      ZT                                                ZT
lim
m!1
           f(u0m(t)jum(t))+kum(t)kH gdt = f(u0(t)ju(t))+ku(t)kH gdt
                                        2
                                            1
                                            0 ()
                                                                             2
                                                                                 1
                                                                                 0 ()
      0                                                 0

TO ESTX SHODIMOSTX \NERGII.
   rASSMOTRIM TEPERX WYRAVENIE
              (u0m(t) ; u0 (t)jum(t) ; u(t)) + kum(t) ; u(t)k2H01() =
= (u0m(t)jum(t)) ; (u0m (t)ju(t)) ; (u0(t)jum(t))+(u0(t)ju(t))+ kum(t)k2H01() ;
                    ;2Re(u(t)jum(t))H01() + ku(t)k2H01():
iSPOLXZUQ SHODIMOSTX \NERGII, IMEEM:
             ZT
   lim Re
  m!1
                  f(u0m(t) ; u0(t)jum(t) ; u(t)) + kum(t) ; u(t)kH gdt = 0
                                                                 2
                                                                     1
                                                                     0 ()
             0



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