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

Abstract

We study local regularity properties of local minimizer of scalar integral functionals of the formwhere the convex integrand F satisfies controlled (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 2estimate for solutions of linear uniformly elliptic equations in divergence form which is optimal with respect to the ellipticity contrast of the coefficients.

Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth

“…There have been parallel contributions from the Soviet school [15,16,25]. In a very nice recent paper, P.Bella and M.Schäffner [2] improved the restriction to…”

“…We define a sequence of test functions η j ∈ C ∞ c (B rj (x 0 )) with the property that 0 ≤ η ≤ 1, η = 1 on B rj+1 (x 0 ) and |∇η j | ≤ C (r j − r j+1 ) 2 . With these choices, the improved Caccioppoli inequality from (5.1) becomes…”