Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth

Abstract

In this work we establish the optimal Lipschitz regularity for non-negative almost minimizers of the one-phase Bernoulli-type functional JG(u, Ω) := F Ω G(|∇u|) + χ{u>0} dx where Ω ⊂ R n is a bounded domain and G : [0, ∞) → [0, ∞) is a Young function with G′ = g satisfying the Lieberman’s classical conditions. Moreover, of independent mathematical interest, we also address a Höder regularity characterization via Campanato-type estimates in the context of Orlicz modulars, which is new for such a class of non-standard growth functionals.