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1、:,.,..,:.,.,.,..2::,,,...,..,,,R2R3.:,,.:ΩR2R3,u∈C4(Ω)C2(Ω¯),u∆u=f(u,∇u)=(1+u2)
2、∇u
3、3,ΩKu.x∈Ω,
4、∇u
5、.0,Ku∂Ω.∇u.:;;.iAbstractThethesisconsistsoffoursections.Thefirstsectionistheintroduction.Itmainlyintroducessom
6、eachievementsanddevelop-mentaboutconvexityoflevelsetsofellipticpartialdifferentialequation.Atthesametime,weintroducewhatwedointhethesis,thatiswediscussitscurvatureestimatesforthelevelsetsofaspecialellipticpartialdifferentialequationinR2andR3.Thesecondsec
7、tionarepreliminaries.Itfocusonthemaximumprincipleandsomerelatedproofs.Furthermore,weintroducesometheorysuchasthedefinitionoftheconvexlevelsetsandthecurvaturematrixofthelevelsetsofafunction.Inthethirdandthefourthsection,wemaketheprioriestimateforthelevel
8、setscurvatureofthesolutionofaspecialellipticpartialdifferentialequation.Inordertoobtaintheconclusion,weuseasubsidiaryfunction,maximumprinciple,andgeometryonboundary.ItisshownthatLetΩbeasmoothboundeddomaininR2orR3andu∈C4(Ω)theellipticequationinΩ,i.e.∆u=f
9、(u,∇u)=(1+u2)
10、∇u
11、3.C2(Ω¯)beasolutionofAssume
12、∇u
13、0inΩ,andthelevelsetsofuarestrictlyconvexwithrespecttonormal∇u.LetKbethecurvatureorGaussiancurvatureofthelevelsetsofu.Thenwehavethefollowingfact:thefunctionKattainsitsminimumontheboundary∂Ω.Keywords:lev
14、elset;convexity;prioriestimates.ii§1§2§3§4.................................................................(1)............................................................(2)...................................................(7)........................
15、....................(11)...............................................................(32)...................................................(34)...................................................................(35)iii§1,,[13],Green.,,Lewis[15]
16、1977.1957Gabriel,Gabrielp-.1982.1990,Lewis[15],Korevaar[9]Caffarelli-Spruck[10]Lewis[15]Gabriel[13]∇u,,:p-.,,([18],[19]).Dirichlet.2008,Longinetti,Jost-Ma-Ou[6]Ma-Ye-Ye[20]Wang-Zhang[1],.,Dirichlet.,,.,R2,R3.
