A remark on local integrability of obstacle problems
Duan Jing · Journal of Natural Science of Heilongjiang University · 2010
Consider A-harmonic equation divA(x,u)=0.Suppose that the operator A satisfies(i) coercivity condition A(x,ξ),ξ≥α|ξ|p-φ1(x);(ii) controllable growth condition |A(x,ξ)|≤β|ξ|p-1+φ2(x) and(iii) homogenity condition A(x,0)=0,where 1pn,0α≤β∞ are non-negative constants,φ1(x)∈Ls/ploc(Ω),φ2(x)∈Ls/(p-1)loc(Ω),1psn.LetKpψ,θ(Ω)={v∈W1,p(Ω):v≥ψ,a.e.Ω,v-θ∈W1,p0(Ω)},ψ be an obstacle defined on Ω valued in R∪{±∞},and θ∈W1,p0(Ω) be the boundary value.By using the inequalities in Sobolev spaces and embedding lemma,it is obtained that if 0≤ψ∈W1,sloc(Ω),then the solution u for the Kpψ,θ-obstacle problem belongs to Ls*loc(Ω) with s*=nsn-s.This result can be regarded as a generalization of the known result due to Gao and Tian.