Calderon-Zygmund Estimates For ω-minimizers Of General Variation Functionals With Double Phase

We study integral functionals involving non-uniformly elliptic operators whose modeled functional is:(?)where Ω(?)R~n denotes a bounded open domain with n≥2 and we shall assume that the numbers p,q and the function a:Ω→[0,∞)satisfy (?)The aim of this paper is to provide Calderon-Zygmund estimate for ω-minimizer of (?)where (?) is the largest possible family of functionals exhibiting non-standard growth conditions and non-uniform ellipticity behaviors.The interest raised on the problem of regularity for ωminimizers is motivated by the fact that,in certain situations,a minimizer of constrained variational problems which are like the obstacle problem or the volume constrained problem can be regarded as an ω-minimizer of an unconstrained one,thereby the treatment is simplified significantly.My strategy is that I find proper covering balls with Vitali’s covering lemma,and then I compare ω-minimiser with a nice function having the Lipschitz regularity over a covering ball.This comparison estimate allows us to control the integral of ω-minimizer over its upper level set.I finally derive the gradient estimate using a truncated function.