Gradient estimates for double phase problems with irregular obstacles

Abstract

An irregular obstacle problem with non-uniformly elliptic operator in divergence form of (p,q)-growth is studied. We find an optimal regularity for such a double phase obstacle problem by essentially proving that the gradient of a solution is as integrable as both the gradient of the assigned obstacle function and the associated nonhomogeneous term in the divergence. Calderón–Zygmund type estimates are also obtained under minimal regularity requirements of the prescribed data.