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On non-local almost minimal sets and an application to the non-local Massari's Problem

Serena Dipierro, Enrico Valdinoci, Riccardo Villa

TL;DR

This work develops a non-local regularity theory for almost minimal sets with respect to the $s$-perimeter, establishing Hölder regularity, a monotonicity formula, and a tangent-cone analysis that yields a sharp bound on the singular set's Hausdorff dimension ($\dim_{\\\\ ext{H}} \Sigma_E \leq n-3$). It also elucidates stickiness phenomena, showing that almost-minimality is stable under external perturbations and that both sticking and non-sticking configurations can arise, including intermediate behaviors. The theory is then applied to a non-local Massari's Problem, proving existence of minimizers for prescribed non-local mean curvature with $H\in L^\infty$ and deriving $C^{1,\alpha}$ regularity of the boundary. Overall, the paper connects fractional perimeter variational problems to practical non-local curvature prescriptions, providing rigorous regularity and structural results with broad implications for non-local capillarity and obstacle-type problems.

Abstract

We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter. In addition, we also discuss stickiness phenomena for non-local almost minimal sets.

On non-local almost minimal sets and an application to the non-local Massari's Problem

TL;DR

This work develops a non-local regularity theory for almost minimal sets with respect to the -perimeter, establishing Hölder regularity, a monotonicity formula, and a tangent-cone analysis that yields a sharp bound on the singular set's Hausdorff dimension (). It also elucidates stickiness phenomena, showing that almost-minimality is stable under external perturbations and that both sticking and non-sticking configurations can arise, including intermediate behaviors. The theory is then applied to a non-local Massari's Problem, proving existence of minimizers for prescribed non-local mean curvature with and deriving regularity of the boundary. Overall, the paper connects fractional perimeter variational problems to practical non-local curvature prescriptions, providing rigorous regularity and structural results with broad implications for non-local capillarity and obstacle-type problems.

Abstract

We consider a fractional Plateau's problem dealing with sets with prescribed non-local mean curvature. This problem can be seen as a non-local counterpart of the classical Massari's Problem. We obtain existence and regularity results, relying on a suitable version of the non-local theory for almost minimal sets. In this framework, the fractional curvature term in the energy functional can be interpreted as a perturbation of the fractional perimeter. In addition, we also discuss stickiness phenomena for non-local almost minimal sets.

Paper Structure

This paper contains 37 sections, 50 theorems, 472 equations, 9 figures.

Table of Contents

  1. Introduction and main results
  2. Regularity of almost minimal sets
  3. Monotonicity formula and size of the singular set
  4. Stickiness for almost minimal sets
  5. The non-local Massari's Problem
  6. Hölder regularity for almost minimal sets and proofs of Lemma \ref{['lemma::almost_minimal_subsupersol']} and Theorem \ref{['th::holder_reg_almost_min']}
  7. Preliminary facts
  8. Uniform Density Estimates
  9. Euler-Lagrange Inequalities
  10. Harnack's Inequality
  11. Improvement of flatness
  12. Proof of Theorem \ref{['th::holder_reg_almost_min']}
  13. Monotonicity formula for almost minimal sets and proofs of Proposition \ref{['prop::char_almost_min_ext']} and Theorem \ref{['th::Phi_monotone']}
  14. Preliminaries for a monotonicity formula
  15. Monotonicity formula and proof of Theorem \ref{['th::Phi_monotone']}
  16. ...and 22 more sections

Key Result

Lemma 1.3

$E$ is an almost minimal set with respect to $\Lambda$ if and only if it satisfies both the $\Lambda$-super-solution property and the $\Lambda$-sub-solution property.

Figures (9)

  • Figure 1: The deformed ball $V_{R,r}$ and the set $\mathbb{E}_{R,r}$ (used in the proof of Lemma \ref{['lemma::aux_EL']}).
  • Figure 2: The sticking half-space.
  • Figure 3: A zoom in of the sticking half-space.
  • Figure 4: Smoothing out the external datum.
  • Figure 5: The non-sticking $\mathcal{C}^0\setminus\mathcal{C}^1$ almost minimizer.
  • ...and 4 more figures

Theorems & Definitions (98)

  • Definition 1.1: Non-local almost minimizers
  • Definition 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Remark 1.8
  • Theorem 1.9: Monotonicity formula
  • Definition 1.10
  • ...and 88 more