Lipschitz Regularity of Almost-Minimizers for Vectorial Alt-Caffarelli Functionals in Orlicz Spaces

Abstract

For a fixed constant λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > 0$$\end{document} and a bounded Lipschitz domain Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^n$$\end{document} with n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional JG(v;Ω):=∫Ω∑i=1mG(|∇vi(x)|)+λχ{|v|>0}(x)dx,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {J}_G(\textbf{v};\Omega ) \,{:}{=}\,\int _\Omega \left( \sum _{i=1}^mG\big (|\nabla v_i(x)|\big ) + \lambda \chi _{\{|\textbf{v}|>0\}}(x)\right) dx , $$\end{document}where v=(v1,⋯,vm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textbf{v} = (v_1, \dots , v_m)$$\end{document} and m∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \in \mathbb {N}$$\end{document}, exhibit optimal Lipschitz continuity on compact subsets of Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, where G is an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {N}$$\end{document}-function satisfying specific growth conditions. Furthermore, we obtain universal gradient estimates for non-negative almost-minimizers in the interior of non-coincidence sets. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the p-Laplacian addressed in Bayrami et al. (2024), and even the scalar case treated in da Silva et al. (2024); Dipierro et al. (2024) and Pellegrino and Teixeira (2024), thereby providing new insights and approaches applicable to a variety of non-linear one or two-phase free boundary problems with non-standard growth.