Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth
João Vítor da Silva, Analía Silva, Hernán Vivas · Discrete and Continuous Dynamical Systems · 2024
In this work we establish the optimal Lipschitz regularity for non-negative almost minimizers of the one-phase Bernoulli-type functional$ \mathcal{J}_{\mathrm{G}}(u, \Omega) \mathrel{\mathop:} = \displaystyle{\int}_\Omega \left(\mathrm{G}(| abla u|)+\chi_{\{u>0\}}\right)\, dx $where $ \Omega \subset \mathbb{R}^n $ is a bounded domain and $ \mathrm{G}: [0, \infty) \to [0, \infty) $ is a Young function with $ \mathrm{G}^{\prime} = 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.