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Regularity for almost minimizers of a one-phase Bernoulli-type functional in Carnot Groups of step two

Fausto Ferrari, Nicoló Forcillo, Enzo Maria Merlino

TL;DR

This work addresses the regularity of almost minimizers for the one-phase Bernoulli-type energy in Carnot groups of step two, extending Euclidean results to a sub-Riemannian setting. The authors adapt the De Silva–Savin dichotomy framework to the subelliptic context by leveraging harmonic replacements, energy- and gradient-estimates, and Morrey–Campanato theory on Carnot groups to control the horizontal gradient. The main achievement is proving intrinsic Lipschitz regularity for nonnegative almost minimizers, with a uniform bound on the horizontal gradient in $B_{1/2}$ and Lipschitz control near the zero set, applicable also to minimizers when $\kappa=0$. This advances understanding of Bernoulli-type free boundary problems in noncommutative, sub-Riemannian geometries and provides a robust intrinsic regularity theory for these variational problems.

Abstract

We prove that nonnegative almost minimizers of the horizontal Bernoulli-type functional $$ J(u,Ω):=\int_Ω\Big(|\nabla_{\mathbb{G}} u(x)|^2+χ_{\{u>0\}}(x)\Big)\,dx$$ are Lipschitz continuous in the intrinsic sense.

Regularity for almost minimizers of a one-phase Bernoulli-type functional in Carnot Groups of step two

TL;DR

This work addresses the regularity of almost minimizers for the one-phase Bernoulli-type energy in Carnot groups of step two, extending Euclidean results to a sub-Riemannian setting. The authors adapt the De Silva–Savin dichotomy framework to the subelliptic context by leveraging harmonic replacements, energy- and gradient-estimates, and Morrey–Campanato theory on Carnot groups to control the horizontal gradient. The main achievement is proving intrinsic Lipschitz regularity for nonnegative almost minimizers, with a uniform bound on the horizontal gradient in and Lipschitz control near the zero set, applicable also to minimizers when . This advances understanding of Bernoulli-type free boundary problems in noncommutative, sub-Riemannian geometries and provides a robust intrinsic regularity theory for these variational problems.

Abstract

We prove that nonnegative almost minimizers of the horizontal Bernoulli-type functional are Lipschitz continuous in the intrinsic sense.
Paper Structure (7 sections, 10 theorems, 210 equations)

This paper contains 7 sections, 10 theorems, 210 equations.

Table of Contents

  1. Introduction and main result
  2. Notation and preliminary results
  3. Carnot Groups
  4. Intrinsic distance and gauge pseudo-distance
  5. Folland-Stein and horizontal Sobolev classes
  6. Dichotomy results
  7. Lipschitz continuity of almost minimizers and proof of Theorem \ref{['theor-Lipsch-contin-alm-minim']}

Key Result

Theorem 1.2

Let $u$ be an almost minimizer for $J$ in $B_1$ with constant $\kappa$ and exponent $\beta$. Then, where $C>0$ is a constant only depending on the homogeneous dimension $Q$, $\kappa,$ and $\beta$. In addition, $u$ is uniformly Lipschitz continuous in a neighborhood of $\left\{u=0\right\}$, namely if $u(0)=0,$ then for some $C>0$, only depending on $Q$, $\kappa$ and $\beta$, and $r_0\in(0,1)$, de

Theorems & Definitions (23)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4: Carnot-Carathéodory distance
  • Proposition 2.5
  • Definition 2.6: Folland-Stein classes
  • Definition 2.7: Horizontal Sobolev spaces
  • Remark 3.1
  • ...and 13 more