Boundary regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals with non-standard growth

2025/12/07 by Pontes, Pedro Fellype, da Silva, João Vitor, Yang, Minbo
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Contact Mechanics and Variational Inequalities #Optimization and Variational Analysis

paper · doi:10.48550/arxiv.2512.06703

Abstract

For a fixed constant λ> 0 and a bounded Lipschitz domain Ω⊂ ℝⁿ with n ≥ 2, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional J_G(\bf v;Ω) \coloneqq ∫_Ω(∑_i=1^mG(|∇ v_i(x)|) + λχ_\|\bf v|gt;0\(x)) dx , where \bf v = (v₁, …, v_m) and m ∈ ℕ, exhibit optimal (up-to-the boundary) Lipschitz continuity, where G is a N-function satisfying specific growth conditions. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the p-Laplacian addressed in \citeBFS24, thereby providing new insights and approaches applicable to a wide class of non-linear one or two-phase free boundary problems with non-standard growth. Our findings remain novel and significant even in the scalar setting and for minimizers of the type considered by Martínez--Wolanski \citeMW08 and da Silva et al. \citedaSSV2024.

Citations

• Lipschitz regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals in Orlicz spaces
• Continuity up to the boundary for minimizers of the one-phase Bernoulli problem
• Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth
• Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian
• A weakly coupled system of p-Laplace type in a heat conduction problem
• Generic properties in free boundary problems
• Lipschitz regularity of almost minimizers in one-phase problems driven by the p-Laplace operator
• Boundary pointwise regularity and applications to the regularity of free boundaries
• Controlling monotonicity of nonlinear operators
• Regularity for Orlicz phase problems
• Regularity of the free boundary for the two-phase Bernoulli problem
• Boundary Regularity for the Free Boundary in the One-phase Problem
• Some remarks on stability of cones for the one-phase free boundary\n problem
• A minimum problem with free boundary in Orlicz spaces
• Boundary regularity of maps with convex potentials

Related

• Lipschitz regularity of weakly coupled vectorial almost-minimizers for Alt-Caffarelli functionals in Orlicz spaces
• Lipschitz regularity of a weakly coupled vectorial almost-minimizers for the p-Laplacian
• Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth
• Lipschitz regularity of almost minimizers in a Bernoulli problem with non-standard growth
• Lipschitz regularity of almost minimizers in one-phase problems driven by the p-Laplace operator