A Quantitative Second Order Sobolev Regularity for(inhomogeneous)Normalized p(·)-Laplace Equations  

作　　者：Yuqing Wang Yuan Zhou 

机构地区：[1]Department of Mathematics,Beihang University,Beijing 102206,P.R.China [2]School of Mathematical Sciences,Beijing Normal University,Beijing 100875,P.R.China

出　　处：《Acta Mathematica Sinica,English Series》2025年第1期99-121,共23页数学学报(英文版)

基　　金：Supported by the National Natural Science Foundation of China(Grant Nos.12025102,12071098,11871088)。

摘　　要：LetΩbe a domain of Rn with n≥2 and p(·)be a local Lipschitz funcion inΩwith 1<p(x)<∞inΩ.We build up an interior quantitative second order Sobolev regularity for the normalized p(·)-Laplace equation−Δ_(p)^(N)(·)u=0 inΩas well as the corresponding inhomogeneous equation−Δ_(p)^(N)(·)u=f inΩwith f∈C^(0)(Ω).In particular,given any viscosity solution u to−Δ_( p(·))^(N)u=0 inΩ,we prove the following:(i)in dimension n=2,for any subdomain U■Ωand anyβ≥0,one has|Du|^(β)Du∈L_(loc)^(2+δ)(U)with a quantitative upper bound,and moreover,the map(x_(1),x_(2))→|Du|^(β)(u_(x1),−u_(x2))is quasiregular in U in the sense that|D[|Du|^(β)Du]|^(2)≤−C detD[|Du|^(β)Du]a.e.inU.(ii)in dimension n≥3,for any subdomain U■Ωwith infU p(x)>1 and supU p(x)<3+2/(n−2),one has D^(2)u∈L_(loc)^(2+δ)(U)with a quantitative upper bound,and also with a pointwise upper bound|D^(2)u|^(2)≤−CΣ_(1≤i<j≤n)[u_(x_(i)x_(j))u_(x_(j)x_(i))−u_(x_(i)x_(i))u_(x_(j)x_(j))]a.e.inU.Here constantsδ>0 and C≥1 are independent of u.These extend the related results obtaind by Adamowicz-H¨ast¨o[Mappings of finite distortion and PDE with nonstandard growth.Int.Math.Res.Not.IMRN,10,1940-1965(2010)]when n=2 andβ=0.

关 键 词：Normalized p(x)-Laplacian strong p(x)-Laplacian second order regularity quasiregular mapping 

分 类 号：O175[理学—数学]