Lipschitz bounds for integral functionals with (p,q)-growth conditions

Abstract We study local regularity properties of local minimizers of scalar integral functionals of the form ℱ ⁢ u: = ∫ Ω F ⁢ (∇ ⁡ u) - f ⁢ u ⁢ d ⁢ x Fu: =_F (u) -fu\, dx where the convex integrand F satisfies controlled (p, q) (p, q) -growth conditions. We establish Lipschitz continuity under sharp assumptions on the forcing term f and improved assumptions on the growth conditions on F with respect to the existing literature. Along the way, we establish an L ∞ L^{} - L 2 L^{2} -estimate for solutions of linear uniformly elliptic equations in divergence form, which is optimal with respect to the ellipticity ratio of the coefficients.