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Regularity of the trace of nonlocal minimal graphs

Serena Dipierro, Ovidiu Savin, Enrico Valdinoci

TL;DR

This work analyzes the boundary trace regularity of nonlocal minimal graphs, where long-range interactions induce stickiness and potential boundary discontinuities. The authors establish $C^{1,\gamma}$ regularity of the trace at stickiness points by first obtaining Lipschitz control of the normal ratio via a localized normal equation and geometric boundary Harnack inequalities, then upgrading to Hölder continuity through a coupled-equation regularity framework. A key consequence is that boundary continuity implies boundary differentiability, supported by a structural continuity result and a cone-triviality argument for nonlocal minimal cones. The findings provide the first regularity results for the boundary trace of nonlocal minimal surfaces and resolve questions about boundary tangents, with potential implications for the analysis of nonlocal geometric variational problems and their boundary behavior.

Abstract

We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~$C^{1,γ}$. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.

Regularity of the trace of nonlocal minimal graphs

TL;DR

This work analyzes the boundary trace regularity of nonlocal minimal graphs, where long-range interactions induce stickiness and potential boundary discontinuities. The authors establish regularity of the trace at stickiness points by first obtaining Lipschitz control of the normal ratio via a localized normal equation and geometric boundary Harnack inequalities, then upgrading to Hölder continuity through a coupled-equation regularity framework. A key consequence is that boundary continuity implies boundary differentiability, supported by a structural continuity result and a cone-triviality argument for nonlocal minimal cones. The findings provide the first regularity results for the boundary trace of nonlocal minimal surfaces and resolve questions about boundary tangents, with potential implications for the analysis of nonlocal geometric variational problems and their boundary behavior.

Abstract

We prove that the trace of nonlocal minimal graphs at points of stickiness is of class~. As a result, we show that boundary continuity implies boundary differentiability for nonlocal minimal graphs.
Paper Structure (13 sections, 14 theorems, 168 equations, 4 figures)

This paper contains 13 sections, 14 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Lipschitz regularity of the trace of nonlocal minimal graphs at points of stickiness
  3. A cutoff version of the equation for the normal
  4. A geometric boundary Harnack inequality
  5. $C^{1,\gamma}$-regularity of the trace of nonlocal minimal graphs at points of stickiness and proof of Theorem \ref{['THM1']}
  6. Weighted averages on annuli
  7. Another geometric boundary Harnack inequality
  8. Regularity theory for coupled equations
  9. $C^{1,\gamma}$-regularity of the trace of nonlocal minimal surfaces at points of stickiness
  10. Differentiability of nonlocal minimal graphs at points of stickiness and proof of Theorem \ref{['2swa1l2e4oj2dn2fl3er4t5e4n2v2']}
  11. Structural continuity at stickiness points
  12. Triviality of nonlocal minimal cones
  13. From the ratio of normal components to the regularity of the boundary trace

Key Result

Theorem 1.1

Let $x_0\in\partial\Omega$ and $X_0:=(x_0,u(x_0))$. Suppose that $\Omega$ is of class $C^{2,1}$ in a neighborhood of $x_0$ and Then, there exists $\rho>0$ such that $u_E(\partial\Omega)\cap B_\rho(X_0)$ is an $(n-1)$-dimensional surface of class $C^{1,\gamma}$ in $\mathbb{R}^{n+1}$.

Figures (4)

  • Figure 1: Sketch illustrating a nonlocal minimal graph "rising" due to far-away data. The figures in this paper serve a purely expository purpose and do not aim to capture the full complexity of nonlocal minimal graphs.
  • Figure 2: Sketch of the exterior normal of a nonlocal minimal graph in the vicinity of a point of stickiness.
  • Figure 3: Sketch of the position of the point $x_\star$ to obtain uniform oscillation bounds for the normal ratio up to the boundary.
  • Figure 4: Sketch illustrating what \ref{['by601widkpw304tyhjb']} wants to avoid.

Theorems & Definitions (27)

  • Theorem 1.1: $C^{1,\gamma}$-regularity of the trace of nonlocal minimal graphs at points of stickiness
  • Theorem 1.2: Differentiability of nonlocal minimal graphs at points of boundary continuity
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 17 more