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

Abstract

We obtain Lipschitz estimates for bounded minimizers of functionals with nonstandard (p, q)-growth satisfying the dimension-independent restriction q < p + 2 with p ≥ 2. This relation improves existing restrictions when p ≤ N − 1, moreover our result is sharp in the range N > p(2 + p) 2 + 1. The standard Lipschitz regularity takes the form W 1,∞ loc − W 1,p loc , whereas we obtain W 1,∞ loc − L ∞ loc regularity estimate and then make use of existing sharp L ∞ loc bounds to obtain the required conclusion.

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

“…In this note, we revisit the question of Lipschitz-regularity for local minimizers of integral functionals of the form (1) w → F (w; Ω) := ˆΩ F (∇w) − f w dx.…”

“…Before we state our main result, we recall a standard notion of local minimality in the context of integral functionals with (p, q)-growth Definition 1. We call u ∈ W 1,1 loc (Ω) a local minimizer of F given in (1) with f ∈ L n loc (Ω) if for every open set Ω ′ ⋐ Ω the following is true:…”

“…Let u ∈ W 1,1 loc (Ω) be a local minimizer of the functional F given in (1) with f ∈ L n,1 (Ω). Then ∇u is locally bounded in Ω.…”

“…is optimal for the estimate in (10) and for n = 3 the exponent is essentially optimal in the sense that the estimate in (10) is in general false for m < 1 2 . While estimate (10) is in some sense optimal, the condition (4) in Theorem 1 is in general not optimal.…”

“…implies that local minimizer of (1) with f ≡ 0 are locally Lipschitz (see also [1] for related discussion). While it is not clear whether (11) is optimal, it strictly improves condition (5) for example in the cases p = 3 and n ≥ 5.…”