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

For a fixed constant λ > 0 λ > 0 and a bounded Lipschitz domain Ω ⊂ Rⁿ Ω ⊂ R n with n ≥ 2 n ≥ 2 , 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)| ) + λ χ _\|v|>0\(x)) dx , J G ( v ; Ω ) : = ∫ Ω ∑ i = 1 m G ( | ∇ v i ( x ) | ) + λ χ { | v | > 0 } ( x ) d x , where v = (v₁, , vₘ) v = ( v 1 , ⋯ , v m ) and m ∈ N m ∈ N , exhibit optimal Lipschitz continuity on compact subsets of Ω Ω , where G is an N N -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.