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