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

Pedro Fellype Pontes, João Vitor da Silva, Minbo Yang

TL;DR

The paper addresses boundary regularity for vector-valued almost-minimizers of Alt-Caffarelli type functionals with nonstandard (Orlicz) growth, establishing optimal Lipschitz continuity up to the boundary. It builds a robust Orlicz-Sobolev framework with $G\in\mathcal{G}(\delta,g_0)$ and introduces non-autonomous operators $\mathfrak{a}$, producing boundary Hölder estimates and energy-decay controls via boundary flattening and $g$-harmonic replacements. The main contributions include proving up-to-boundary Lipschitz regularity for $(\kappa,\beta)$-almost-minimizers, deriving boundary Hölder regularity for the minimizers and their gradients, and extending vectorial nonstandard-growth regularity results to one- and two-phase free boundary problems in the Orlicz setting. These results broaden the Alt-Caffarelli framework to nonstandard growth, providing tools for analysis of Bernoulli-type free boundary problems in Orlicz spaces and highlighting the applicability to a wide class of nonlinear variational problems.

Abstract

For a fixed constant $λ> 0$ and a bounded Lipschitz domain $Ω\subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional \[ \mathcal{J}_G({\bf v};Ω) \coloneqq \int_Ω\left(\sum_{i=1}^mG\big(|\nabla v_i(x)|\big) + λχ_{\{|{\bf v}|>0\}}(x)\right) dx , \] where ${\bf v} = (v_1, \dots, v_m)$ and $m \in \mathbb{N}$, exhibit optimal (up-to-the boundary) Lipschitz continuity, where $G$ is a $\mathcal{N}$-function satisfying specific growth conditions. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the $p$-Laplacian addressed in \cite{BFS24}, 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 \cite{MW08} and da Silva \textit{et al.} \cite{daSSV2024}.

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

TL;DR

The paper addresses boundary regularity for vector-valued almost-minimizers of Alt-Caffarelli type functionals with nonstandard (Orlicz) growth, establishing optimal Lipschitz continuity up to the boundary. It builds a robust Orlicz-Sobolev framework with and introduces non-autonomous operators , producing boundary Hölder estimates and energy-decay controls via boundary flattening and -harmonic replacements. The main contributions include proving up-to-boundary Lipschitz regularity for -almost-minimizers, deriving boundary Hölder regularity for the minimizers and their gradients, and extending vectorial nonstandard-growth regularity results to one- and two-phase free boundary problems in the Orlicz setting. These results broaden the Alt-Caffarelli framework to nonstandard growth, providing tools for analysis of Bernoulli-type free boundary problems in Orlicz spaces and highlighting the applicability to a wide class of nonlinear variational problems.

Abstract

For a fixed constant and a bounded Lipschitz domain with , we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type functional where and , exhibit optimal (up-to-the boundary) Lipschitz continuity, where is a -function satisfying specific growth conditions. Our work extends the recent regularity results for weakly coupled vectorial almost-minimizers for the -Laplacian addressed in \cite{BFS24}, 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 \cite{MW08} and da Silva \textit{et al.} \cite{daSSV2024}.

Paper Structure

This paper contains 6 sections, 17 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. State-of-the-art and some motivational related results
  3. On the Orlicz and Orlicz-Sobolev spaces
  4. Auxiliary estimates for non-autonomous operator
  5. Regularity near the boundary
  6. Boundary Lipschitz regularity

Key Result

Theorem 1.1

Let $\Omega$ be a $C^{1,\alpha}$-domain of $\mathbb{R}^n$, and $G$ be an $\mathcal{N}$-function such that $G \in \mathcal{G}(\delta,g_0)$, with $\delta>1$. Let $\mathbf{u}:\Omega \to \mathbb{R}^m$ be a $(\kappa,\beta)$-almost-minimizer of $\mathcal{J}_G$ in $\Omega$, with constant $\kappa \le \kappa

Theorems & Definitions (37)

  • Definition 1.1: $\mathcal{N}$-function
  • Definition 1.2: Non-degenerate class
  • Theorem 1.1: Lipschitz regularity up to the boundary
  • Corollary 1.1: Boundary Lipschitz regularity for minimizers
  • Definition 2.1: Orlicz spaces
  • Definition 2.2: $\Delta_2$ and $\nabla_2-$condition
  • Lemma 2.1
  • Lemma 2.2: see BC
  • Lemma 2.3
  • proof
  • ...and 27 more