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<p>In this note, we present some regularity results for almost minimizers of Bernoulli-type functionals in Carnot groups of step two, recently obtained in (F. Ferrari et al. Regularity of almost-minimizer of a one-phase Bernoulli-type functional in Carnot Group of step two, in press in Calc. Var. Partial Differential Equations (2025) arXiv:2311.02975v2; F. Ferrari and E. M. Merlino, Lipschitz regularity for almost minimizers of a one-phase p-Bernoulli-type functional in Carnot Groups of step two, submitted, available on arXiv:2407.00084). We focus mostly on the functional with quadratic growth, treated in (Ferrari, F., Forcillo, N., Merlino, E.M.: Regularity of almost-minimizer of a one-phase Bernoulli-type functional in Carnot Group of step two, in press in Calc. Var. Partial Differential Equations (2025), available on <a href="http://arxiv.org/abs/2311.02975v2">arXiv:2311.02975v2</a>), and describe what are the main issues and techniques used in order to establish locally Lipschitz regularity for almost minimizers.</p>

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A note on regularity of almost minimizers of horizontal one-phase Bernoulli problem

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Enzo Maria Merlino

摘要

In this note, we present some regularity results for almost minimizers of Bernoulli-type functionals in Carnot groups of step two, recently obtained in (F. Ferrari et al. Regularity of almost-minimizer of a one-phase Bernoulli-type functional in Carnot Group of step two, in press in Calc. Var. Partial Differential Equations (2025) arXiv:2311.02975v2; F. Ferrari and E. M. Merlino, Lipschitz regularity for almost minimizers of a one-phase p-Bernoulli-type functional in Carnot Groups of step two, submitted, available on arXiv:2407.00084). We focus mostly on the functional with quadratic growth, treated in (Ferrari, F., Forcillo, N., Merlino, E.M.: Regularity of almost-minimizer of a one-phase Bernoulli-type functional in Carnot Group of step two, in press in Calc. Var. Partial Differential Equations (2025), available on arXiv:2311.02975v2), and describe what are the main issues and techniques used in order to establish locally Lipschitz regularity for almost minimizers.