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.