YuDSalmanovTransactionsofNASofAzerbaijankFkL2ku08kL2i20Cp - PDF document

1 168 [Yu.D.Salmanov]TransactionsofNASofAz
168 [Yu.D.Salmanov]TransactionsofNASofAzerbaijan+kFkL2( )ku�kL2( )iCp D(u�)kkWr2( )+kFk2+Ckkpp:ConsequentlyD1=2(u�)CkkWr2( )+kkp 2p+kFk2;kukp 2pCkkWr2( )+kkp 2p+kF2k;wherethepositiveconstantisindepentof;F.Byvirtueof(5)wehaveku�kWr2( )D1=2(u�),sinceu�2Wr2( ),soku�kWr2( )CkkWr2( )+kkp 2p+kFk2:ThuskukWr2( )CkkWr2( )+kkp 2p+kFk2:(12)Theorem2.Letcoecientsakl(x)ofequation(8)haveboundedpartialderivatesoforderm(mr)inclusivelyindomain .a(s)kl(x)M;(s=(s1;:::;sn);jsjm):(13)

2 Then,atconditions(6)andF(x)2L2( )general
Then,atconditions(6)andF(x)2L2( )generalizedsolutionofboundaryproblem(9)belongstoweightedclassWr+m2;2�p 2pn( )(1p2),besides,forittheinequalityisvalid(see(10)).kukWr+m2;2�p 2pn( )kukL2( )+0@Xjkj=r+m 2�p 2pnu(k) 2L2( )1A1 2max�K;Kp�1
:(14)Proof.Wexarbitrarypointx0=�x01;:::;x0n
2 andlet;1betwoballswithcentersatthepointx0whoseradiirespectivelyequalto1 4(x0)and1 2(x0)where(x)isdistancefromthepointx2 totheboundary�.Wegiveentirenonnegativevectors=(s1;:::;sn)(sj0),denoteby
shf(x)s�mixeddierenceoffunctionf(x)atthepointxwithintervalh:
sf(x)=
shf(x)=
s1he1
s2he2:::
snhenf(x1;x2;:::xn);wh

3 ere
siheiisanoperatorofsi-folddi
ere
siheiisanoperatorofsi-folddierencebyvariablexi.Let(x)2Wr2(0).Sincefunction(x)hassupportin0,thenatsucientlysmallhfrom(8)canobtaintheequality(seeforexample[4;6]):Z0Xjkjrjljr
sakl(x)u(k)(x)(l)(x)dx+Z0
sa(x)ju(x)jp�2u(x) 170 [Yu.D.Salmanov]TransactionsofNASofAzerbaijanLeavingalltermsoftheconsideredmultiplesumcorrespondingtojkj=jlj=rintheleftpartof(17),transferringremaindertermstotherightpartof(17),andthenusingtheinequality(6)totheleftpart,wehave0X20Z0Xjkj=r
su(k)(x) hjsj!2(x)dxJZ0Xjkjrjljrakl(x)
su(k)(x) hjsj
su(l)(x) hjsj&

4 #17;(x)dx==�X0Jk;l;;�J1+
#17;(x)dx==�X0Jk;l;;�J1+J2;(18)whereJk;l;;=CsClZ0
akl[x+h(s�)] hjj
s�u(k)(x) hjs�j
su(l�)(x) hjsj()(x)dx;J1=Z0
s(a(x)ju(x)jp�2u(x)) hjsj
su(x) hjsj(x)dx;J2=Z0
sF(x) hjsj
su(x) hjsj(x)dx;k;l;;areentirenon-negativevectors,forwhichjkjr;jljr;jkj+jlj2r;0s;0l.Let'sestimateintegralsintherightpartof(18).Assumingasyetthatu(x)2Wr+jsj�12(0).1)if=0andjkj=r(thenjljr),weobtainjJk;l;0;jCZ0
su(k)(x) hjsj
su

5 (l�)(x) hjsj
(l�)(x) hjsj()(x)dxC0@Z0
su(k)(x) hjsj2dx1A1 20B@Z
su(l�)(x) hjsj2dx1CA1 2CX u(l�+s) L2(0)CXkukWr+jsj�12(0):(19)Byvirtueoftheconditionjakl(x)jM(x2 ),therstrelationofthischainofinequalitiesisvalid,thesecondisknown,inthethirditisusedthecondition4)offunction(x)andthefact,thatinthepresentcasejs+l�jr+jsj�1.Thereforewecanusetheestimateofdierencerelationbycorrespondingderivative(see[6]). Transactionsof

6 NASofAzerbaijan [Onsmooth.ofgeneral.solu
NASofAzerbaijan [Onsmooth.ofgeneral.solut.ofellip.equat.]1712)If=0;jlj=r(thenjkjr),sojkj+js�jr+jsj�1;thereforejJk;l;;0jC0B@Z
s�u(k)(x) hjs�j!2dx1CA1 20@Z0
su(l)(x) hjsj!2(x)dx1A1 2kukWr+jsj�12(0)X:(20)3)Intherestcases(i.e.whenjkjr;jljr)wehavejs�j+jkjr+jsj�1;jsj+jl�jr+jsj�1.Estimatingdierencequotientfora()kl(x)(sincebyconditionja()kl(x)jM;jjm),andfunction()(x)withconstantweobtainjJk;l;;jC0B@Z
s�u(k)(x) hjs�j!2dx1CA1 20B@Z
su(l�)(x) hjsj!2dx1CA1 2kuk2Wr+jsj�12(

7 0):(21)Byvirtueof(19)-(21)weobtain&
0):(21)Byvirtueof(19)-(21)weobtain�X0Jk;l;;C1kukWr+jsj�12(0)X+C2kuk2Wr+jsj�12(0);(22)whereC1;C2isindependentofu.Usingtheformulaanalogoustoformulaofintegrationbyparts,byvirtueofthenitenessof(x)in0forintegralJ1from(18)wehavejJ1j=R0
sh hjsj�a(x)ju(x)jp�1signu(x)

shu(x) hjsj(x)dx==R0a(x)ju(x)jp�1signu(x)
s�h hjsj
shu(x) hjsj(x)dxkukp�1Lp(0) 
shu hjsj(s) L2(0)8:kukp�1Lp(0)X+kukp�1Lp(0)kukWr+jsj�12

8 (0);jsj=r;kukp�1Lp(0)kukWr+
(0);jsj=r;kukp�1Lp(0)kukWr+jsj�12(0);jsjr(23)ByanalogyforintegralJ2from(18)wehavejJ2j8:kFkL2(0)X+kFkL2(0)kukWr+jsj�12(0);jsj=r;kFkL2(0)kukWr+jsj�12(0);jsjr;(24) TransactionsofNASofAzerbaijan [Onsmooth.ofgeneral.solut.ofellip.equat.]173InthefollowingitisusedthefollowingtheoremofTruazi[6]:Forarbitraryboundedmeasurabledomain RnitholdsR s�x0
kfkLp((x0))pdx0~R s+n p(x)jf(x)jpdx;where1p1;sisanarbitraryrealnumber,�x0
isanyballwithcenterx0,stronglylyinginside ;isanequivalencesign.Wemultiplybothpatsof(27)by�x0
�p�1 pnandthenwesquareitandintegratebyx

9 02 ,thenweapplytoeachtermoftheobtainedin
02 ,thenweapplytoeachtermoftheobtainedinequalitytheTruazitheoremandifweraiseresulttothepower1 2,thenwehavekukWr+m2;2�p 2pn( )kukWr2( )+kukp�1Lp( )+kFkL2( ):Thisinequalitytogetherwith(10)proves(14)(1p2).References[1].BesovO.V.,IlinV.P.,Nikol'skiS.M.Theintegralrepresentationsoffunc-tionsandimbeddingtheorems.M.:"Nauka",1975,p.480.(Russian)[2].Nikol'skiS.M.Approximationoffunctionsofseveralvariablesandembed-dingtheorems.M.:"Nauka",1977,p.455.(Russian)[3].SalmanovYu.D.Investigationofspecialproblemsoftheoryofdierentialequations.Baku,Elm,1974,pp.171-184.(Russian)[4].SalmanovYu.D.Doctor'sdissertation.Baku,1999.(Russian)[5].Nikol'skiS.M.Avariatio

10 nalproblemforanequationofelliptictypewit
nalproblemforanequationofelliptictypewithdegenerationontheboundary.TrudyMIAN,1979,v.150,pp.212-238.(Russian)[6].LizorskinP.I.,Nikols'kiS.M.Coercivepropertiesofellipticequationswithdegeneracy.Variationalmethod.TrudyMIAN,1981,v.1957,pp.90-118.(Russian)[7].SalmanovYu.D.Theboundaryvalueproblemofrsttypeforsomenon-lineardierentialequationswithdegenerationonmanifoldsofarbitrarydimension.Di.uravn.,1999,v.35,No12,pp.1677-1683.(Russian) 174 [Yu.D.Salmanov]TransactionsofNASofAzerbaijanYusifD.SalmanovAzerbaijanStatePedagogicalUniversity.34,U.Hajibeyovstr.,AZ1004,Baku,Azerbaijan.Tel.:(99412)4933369(o.)RecievedSeptember22,2003;RevisedFebruary24,2004.TranslatedbyMame