Global integrability related to anisotropic operators

We consider boundary value problems of the form{∑i=1nDi(ai(x,Du(x)))=∑i=1nDifi(x), in Ω,u(x)=u⁎(x), on ∂Ω, where ai:Ω×Rn→Rai:Ω×Rn→R, i=1,2,⋯,ni=1,2,⋯,n, are Carathéodory functions satisfying∑i=1n|ai(x,z)|pi′≤c1∑i=1n|zi|pi+g1(x), andc2∑i=1n|zi|pi−g2(x)≤∑i=1nai(x,z)zi for some positive constants c1c1, c2c2 and some functions g1g1, g2g2. We also consider minimizers u∈u⁎+W01,(pi)(Ω) of the integral functionalI(u)=∫Ωf(x,Du(x))dx, where the integrand f(x,z):Ω×Rn→[0,+∞)f(x,z):Ω×Rn→[0,+∞) satisfiesc3∑i=1n(∑j=1n|zj|pj)pi−2pi|zi|2−g3(x)≤f(x,z)≤c4∑i=1n(∑j=1n|zj|pj)pi−2pi|zi|2+g4(x) for some positive constants c3c3, c4c4 and some functions g3g3, g4g4. We show, by a different method from the classical ones, that higher integrability of the boundary datum u⁎u⁎ forces u to have higher integrability as well. Similar results are also obtained for obstacle problems.