Lipschitz bounds for nonuniformly elliptic integral functionals in the plane

We study local regularity properties of local minimizer of scalar integral functionals with controlled (p, q) -growth in the two-dimensional plane. We establish Lipschitz continuity for local minimizer under the condition 1<p q< with q<3p which improve upon the classical results valid in the regime q<2p. Along the way, we establish an L^-L²-estimate for solutions of linear uniformly elliptic equations in the plane which is optimal with respect to the ellipticity contrast of the coefficients.