Lipschitz bounds for nonuniformly elliptic integral functionals in the plane

Abstract

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\leq q>\infty 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^\infty - L 2 L^2 -estimate for solutions of linear uniformly elliptic equations in the plane which is optimal with respect to the ellipticity contrast of the coefficients.