Уравнения математической физики (пространства Соболева). Салехов Л.Г - 14 стр.

PROSTRANSTW WYTEKAET, ^TO H s (Rn)]0  S 0 (Rn).
  f) pROSTRANSTWO H s (Rn)]0 DUALXNOE K PROSTRANSTWU H s(Rn) IZO
                                        ,                                    ,      -

METRI^ESKI IZOMORFNO PROSTRANSTWU H ;s (Rn) TO ESTX H s (Rn)]0 =
                                                             ,

H ;s (Rn).
   dOKAZATELXSTWO.            pUSTX u 2 H s (Rn)]0 . pOSKOLXKU H s (Rn)]0 
S 0 (Rn), TO u 2 S 0 (Rn).
   rASSMOTRIM MAKSIMUM WYRAVENIQ
                                 ((1 + j j2);s=2u^j )
KOGDA PROBEGAET MNOVESTWO fk kL 6 1g. |TOT MAKSIMUM RAWEN
                                                 2


                             k(1 + j j2);s=2 u^kL = kuk;s :
                                                     2


tAK KAK (1+ j j2)s=2 2 M (Rn), TO MOVNO S^ITATX, ^TO = (1+ j j2)s=2'^ ,
GDE ' 2 S (Rn), PRI^EM USLOWIE k kL 6 1 \KWIWALENTNO k'ks 6 1.
                                                     2

sLEDOWATELXNO, ISPOLXZUQ TEOREMU pARSEWALQ, POLU^IM
   ((1 + j j2);s=2u^j ) = ((1 + j j2);s=2u^j(1 + j j2)s=2'^ ) = (^uj'^ ) = (uj'):
dALEE IMEEM:
                 j(uj')j 6 kukH s(Rn)]0 k'ks 6 kukH s(Rn)]0 :
oTKUDA (1 + j j2);s=2 u^ 2 L2(Rn), TO ESTX u 2 H ;s (Rn) I
                            kuk;s 6 kukH s(Rn)]0 :                           ()
  oBRATNO, PUSTX u 2 H ;s (Rn). tAK KAK kukH ;s(Rn)]0 = kmax   'ks61
                                                                      j(uj')j, TO
DOSTATO^NO OCENITX WYRAVENIE
                         ((1 + j j2);s=2u^j(1 + j j2)s=2'^ )
PRI USLOWII, ^TO ' 2 S (Rn) k'ks 6 1.
  iMEEM:
                j(uj')j = j((1 + j j2);s=2 u^j(1 + j j2)s=2'^ )j 6
             k(1 + j j2);s=2u^kL k(1 + j j2)s=2'^kL = kuk;s k'ks 
                                    2                    2


OTKUDA u 2 H s (Rn)]0 I KROME TOGO
                                 kukH s(Rn)]0 6 kuk;s :                         ()
                                            14