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Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth

Chiara Leone, Giovanni Scilla, Francesco Solombrino, Anna Verde

TL;DR

We address local Lipschitz regularity for scalar almost-minimizers of Alt-Caffarelli-type functionals with generalized Orlicz growth, capturing nonstandard growth and space-inhomogeneity through $\varphi$.The approach builds a robust non-autonomous Orlicz-Sobolev framework, constructs a regularized $\widetilde{\varphi}$, and uses comparison with $\widetilde{\varphi}$-harmonic replacements together with blow-up arguments to obtain $C^{0,\alpha}$ and $C^{1,\alpha}$-type results away from the free boundary, plus sublinear behavior near the free boundary.Key contributions include a unified Lipschitz regularity theory for almost-minimizers under nonstandard growth (including perturbed Orlicz, variable exponent, and double-phase energies), with a versatile toolkit for autonomous and non-autonomous settings and potential extension to vector-valued problems.The results pave the way for a deeper understanding of free boundary regularity in nonstandard growth settings and provide stable regularity estimates that can inform numerical methods for related phase-field problems.

Abstract

The optimal local Lipschitz regularity for scalar almost-minimizers of Alt-Caffarelli-type functionals $$ \mathcal{F}({v}; Ω) = \int_Ω\varphi(x,\left|\nabla v(x) \right|)+ λχ_{\{{v} >0\}} (x) \, \mathrm{d}x\,, $$ with growth function $\varphi$ a generalized Orlicz function, is established.

Lipschitz regularity of almost-minimizers in one-phase problems with generalized Orlicz growth

TL;DR

We address local Lipschitz regularity for scalar almost-minimizers of Alt-Caffarelli-type functionals with generalized Orlicz growth, capturing nonstandard growth and space-inhomogeneity through $\varphi$.The approach builds a robust non-autonomous Orlicz-Sobolev framework, constructs a regularized $\widetilde{\varphi}$, and uses comparison with $\widetilde{\varphi}$-harmonic replacements together with blow-up arguments to obtain $C^{0,\alpha}$ and $C^{1,\alpha}$-type results away from the free boundary, plus sublinear behavior near the free boundary.Key contributions include a unified Lipschitz regularity theory for almost-minimizers under nonstandard growth (including perturbed Orlicz, variable exponent, and double-phase energies), with a versatile toolkit for autonomous and non-autonomous settings and potential extension to vector-valued problems.The results pave the way for a deeper understanding of free boundary regularity in nonstandard growth settings and provide stable regularity estimates that can inform numerical methods for related phase-field problems.

Abstract

The optimal local Lipschitz regularity for scalar almost-minimizers of Alt-Caffarelli-type functionals with growth function a generalized Orlicz function, is established.

Paper Structure

This paper contains 11 sections, 22 theorems, 151 equations.

Table of Contents

  1. Introduction and the main results
  2. Basic notation and preliminaries
  3. Generalized $\Phi$-functions and Orlicz spaces
  4. Regularized Orlicz function
  5. Some technical lemmas
  6. Local Lipschitz regularity of almost minimizers
  7. Regularity estimates for autonomous problems
  8. Preliminary regularity results for almost-minimizers and comparison estimates
  9. Local Hölder continuity
  10. Proof of the local Lipschitz continuity
  11. Acknowledgements

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^d$ be a bounded open set. Let $\varphi \in \Phi_{\rm{c}}(\Omega)$, $\varphi(x,\cdot)\in C^1([0,\infty))$ be satisfying va1, and such that $\varphi_t$ comply with azero, inc and dec for some $1<p\leq q$. Let ${u} :\Omega \to \mathbb{R}$ be an almost-minimizer of $\mathcal

Theorems & Definitions (41)

  • Definition 1.1
  • Theorem 1.1: Interior regularity
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • ...and 31 more