[Paper Review] Regularity for non-local almost minimal boundaries and applications

Michèle Caputo, Nestor Guillen|arXiv (Cornell University)|Mar 12, 2010

Nonlinear Partial Differential Equations10 references19 citations

TL;DR

This paper establishes a non-local version of the Almgren-De Giorgi-Tamanini regularity theory for almost minimal boundaries by proving that flat non-local almost minimal sets are $C^{1, ho}$ regular, extending classical results to fractional Sobolev energies. The key contribution is a full regularity theory for non-local almost minimal boundaries, including $C^{1, ho}$ regularity near flat points and an $n-2$ dimensional singular set estimate, via novel variational inequalities and monotonicity formulas.

ABSTRACT

We introduce a notion of non-local almost minimal boundaries similar to that introduced by Almgren in geometric measure theory. Extending methods developed recently for non-local minimal surfaces we prove that flat non-local almost minimal boundaries are smooth. This can be viewed as a non-local version of the Almgren-De Giorgi-Tamanini regularity theory. The main result has several applications, among these $C^{1,α}$ regularity for sets with prescribed non-local mean curvature in $L^p$ and regularity of solutions to non-local obstacle problems.

Motivation & Objective

• To extend the Almgren-De Giorgi-Tamanini regularity theory to non-local almost minimal boundaries defined by fractional Sobolev energies.
• To establish $C^{1, ho}$ regularity for sets that are almost minimal with respect to the $H^{s/2}$ energy functional.
• To prove that the singular set of such boundaries has Hausdorff dimension at most $n-2$, mirroring the classical result.
• To develop variational inequalities as substitutes for the Euler-Lagrange equation in the non-local almost minimal setting.
• To provide a foundation for applications in non-local mean curvature problems and obstacle problems.

Proposed method

• Introduce a notion of $(J_s, \rho, \delta)$-minimality for non-local energy functionals, generalizing classical almost minimal sets.
• Prove uniform volume density estimates using a non-local extension of De Giorgi’s differential inequality, complementing discrete iteration methods.
• Establish a variational inequality (Theorem 5.15) that serves as a substitute for the minimal surface equation in the almost minimal case.
• Derive a partial Harnack estimate (Theorem 5.18) to enable an improvement of flatness via blow-up and compactness arguments.
• Prove a monotonicity formula for the energy functional $\Phi_E(r)$, showing it is non-decreasing and vanishing only for homogeneous functions.
• Apply blow-up analysis and dimension reduction techniques to conclude that the singular set has Hausdorff dimension at most $n-2$.

Experimental results

Research questions

• RQ1 Under what conditions is a non-local almost minimal boundary $C^{1,\rho}$ regular near a flat point?
• RQ2 Can the classical Almgren-De Giorgi-Tamanini regularity theory be extended to non-local energy functionals?
• RQ3 What variational inequality characterizes non-local almost minimal sets in the absence of a classical Euler-Lagrange equation?
• RQ4 How does the singular set of a non-local almost minimal boundary behave in terms of Hausdorff dimension?
• RQ5 Can the monotonicity formula for the energy functional be used to classify blow-up limits and prove regularity?

Key findings

• If a set $E$ is $(J_s, \rho, \delta)$-minimal in $B_1$ and $\delta$-flat with $\delta \leq \delta_0(n,s,\rho)$, then $\partial E$ is $C^{1}$ in $B_{1/2}$.
• The set of points where $\partial E$ is not $C^1$ has Hausdorff dimension at most $n-2$.
• The energy functional $\Phi_E(r)$ is non-decreasing in $r$, and $\Phi_E'(r) \geq \frac{1}{r^{n-s}} \int_{\partial B_r^+} z^a (\hat{u}_\nu)^2 dX$, implying monotonicity.
• If $\Phi_E(0) < \Phi_H + \delta_0$ for some $\delta_0 > 0$, then $\partial E$ is $C^{1,\hat{\rho}}$ near the origin.
• Blow-up limits of almost minimal sets are minimal cones, and if the cone is not a half-space, the energy gap $\Phi_C \geq \Phi_H + \delta_0$ holds.
• For the non-local obstacle problem, $\partial E$ is $C^{1,\alpha}$ in a neighborhood of the obstacle for some $\alpha < s$.