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

Source abstract

Abstract For a fixed constant $$\lambda > 0$$ λ > 0 and a bounded Lipschitz domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n with $$n \ge 2$$ n ≥ 2 , we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional $$\mathcal {J}_G(\textbf{v};\Omega ) \,{:}{=}\,\int _\Omega \left( \sum _{i=1}^mG\big (|\nabla v_i(x)|\big ) + \lambda \chi _{\{|\textbf{v}|>0\}}(x)\right) dx ,$$ J G ( v ; Ω ) : = ∫ Ω ∑ i = 1 m G ( | ∇ v i ( x ) | ) + λ χ { | v | > 0 } ( x ) d x , where $$\textbf{v} = (v_1, \dots , v_m)$$ v = ( v 1 , ⋯ , v m ) and $$m \in \mathbb {N}$$ m ∈ N , exhibit optimal Lipschitz continuity on compact subsets of $$\Omega$$ Ω , where G is an $$\mathcal {N}$$ 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.