The α‐Fractional Gradient Estimates for Solutions to the Borderline Case of Double‐Phase Elliptic Problems

ABSTRACT We study the gradient regularity of weak solutions to a class of elliptic equations driven by the borderline case of double‐phase operators. Due to the non‐uniform ellipticity and the transition from polynomial to a logarithmic‐polynomial type, the problem at hand poses challenging tasks. Our work is motivated by recent contributions to the existence and preliminary regularity results of logarithmic double‐phase problems, as well as the nonlinear problems with unbalanced growth, which have sparked interest in the calculus of variations. In this paper, a generalized gradient estimate for weak solutions to this problem is provided in the setting of weighted Lorentz spaces. Furthermore, with the relation between fractional derivatives and fractional Sobolev spaces (a function in fractional Sobolev spaces is related to its fractional maximal operators in some specific Lebesgue space), through the , we can derive the global ‐fractional gradient estimates for solutions to this problem under suitable assumptions on the ground domain. Our work in this paper is not only a contribution to the study of regularity theory in problems exhibiting non‐standard growth, but also brings a connection with numerous applications in real‐life models, such as composite material mechanics, nonlinear elasticity, non‐Newtonian fluids, image processing, and so forth.