障碍问题局部可积性的一个注记

考虑A-调和方程divA(x,u)=0,设算子A满足:(i)强制性条件A(x,ξ),ξ≥α|ξ|p-φ1(x);(ii)控制增长条件|A(x,ξ)|≤β|ξ|p-1+φ2(x);(iii)齐次性条件A(x,0)=0,其中1pn,0α≤β∞是非负常数,φ1(x)∈Llso/cp(Ω),φ2(x)∈Lslo/c(p-1)(Ω),1psn。设Kψp,θ(Ω)={v∈W1,p(Ω):v≥ψ,a.e.Ω,v-θ∈W01,p(Ω)},ψ为定义于Ω取值于R∪{±∞}的障碍函数,θ∈W01,p(Ω)为边值。利用Sobolev空间的不等式及嵌入引理,得到了如下局部可积性结果:若0≤ψ∈Wl1o,cs(Ω),则Kψp,θ-障碍问题的解u∈Llso*c(Ω),s*=nn-ss。本结果可看成是高红亚,田会英的结果的推广。

A remark on local integrability of obstacle problems

Consider A-harmonic equation divA (x,△u) = 0. Suppose that the operator A satisfies (i) coercivity condition ＜A(x,ζ) ,ζ＞≥α|ζ|~p-φ_1(x);(ii) controllable growth condition |A(x,ζ)|≤β|ζ|~(p-1)+φ_2(x) and (iii) homogenity condition A (x,0) = 0, where 1 ＜p ＜ n, 0 ＜α≤β＜∞ are non-negative constants, φ_1(x)∈L_(loc)~(s/p) (Ω),ψ_2(x)∈L_(loc)~(s/(p-1)) (Ω),1＜p＜s＜n. LetK_(ψ,θ)~p(Ω)={v∈W~(1,p)(Ω):v≥ψ,a.e.Ω,v-θ∈W_0~(1,p) (Ω)},be an obstacle defined on Ω valued in R∪{±∞}, and θ∈W_0~(1,p)(Ω) be the boundary value. By using the inequalities in Sobolev spaces and embedding lemma,it is obtained that if 0≤ψ∈W_(loc)~(1,s) (Ω), then the solution u for the K_(ψ,θ)~p-obstacle problem belongs to L_(loc)~(s*)(Ω) with s~*=ns/n-s. This result can be regarded as a generalization of the known result due to Gao and Tian.