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

Abstract

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 \mathcal{F}[u]:=\int_{\Omega}F(\nabla 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^{\infty}} - 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.