一类p(x)LAPLACE方程弱解的正则性.pdf

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1、
UDC:p(x)−Laplace$%!"#4%&'()*+,(-./012356789:201156=>9:;11<2011;12<$%!"#()?@A)=>BCDEFGHI2011J11KDissertationSubmittedtoHebeiUiversityofTechnologyforTheMasterDegreeofScienceinAppliedMathematicsTHEREGULARITYOFWEEKSOLUTIONFORp(x)

2、−LAPLACEEQUATIONbyChenWeiSupervisor:Prof.LiuHuiZhaoNovember2011ùúûüýþÿþ
ªNp(x)−Laplacep(x)-Laplace(p(x)>1)−div(

3、∇u

4、p(x)−2∇u)=f(x,u,∇u)x∈Ωp(x)∈C(Ω)f(x,u,∇u)!"

5、f(x,u,η)

6、≤h(x)+a(x)

7、u

8、r(x)−1+b(x)

9、η

10、α(x),∀x∈Ω,u∈R,η∈Rn#$()*%&'/012!"Caccioppol

11、i+,-.3*89%&'[35]4567p(x)-Laplace!":;<=>*;<%&'!"67p(x)-Laplace?@/012B(Ω,M,γ,γ,δ,1)DEm(x)1qABC[31]45p(x)-89:Laplace%&'!"H¨olderFG=H*3:KD45IJ[28]45=LMNOYPYQYYUÝRp(x)−Laplace¯H¨olderSTéZVWXYi[p(x)−Laplace]^_`abcdTHEREGULARITYOFWE

12、EKSOLUTIONFORp(x)−LAPLACEEQUATIONABSTRACTInthisthesis,weconsidertheregularityofweeksolutionforthefollow-ingp(x)−Laplaceequation−div(

13、∇u

14、p(x)−2∇u)=f(x,u,∇u)x∈Ωherep(x)∈C(Ω)andf(x,u,∇u)satisfythecondition

15、f(x,u,η)

16、≤h(x)+a(x)

17、u

18、r(x)−1+b(x)

19、η

20、α(x),∀x∈Ω,u∈R,η∈RnFirstly,

21、underthecertainconditions,weobtaintheCaccioppolitypeinequalityfortheweeksolutionofp(x)-Laplaceequation.Usingthere-sultestablishedin[35],weobtainthattheweeksolutionsofp(x)−Laplaceequationarelocallybounded.Secondly,underthesomeassumptions,weprovethatiftheweeksolution

22、sofp(x)-LaplaceequationisboundedthenitmustbelongtoB(Ω,M,γ,γ,δ,1)andthereforeitisalsolocalH¨oldercontinuitym(x)1qbythetheoremobtainedin[31].KEYWORDS:p(x)−Laplacespaces,Localboundedness,localH¨oldercontinuity,Non-satandardgrowthcondition.iiùúûüýþÿþ
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