[Paper Review] Almost minimizers for certain fractional variational problems

Seongmin Jeon, Arshak Petrosyan|arXiv (Cornell University)|May 28, 2019

Nonlinear Partial Differential Equations7 references5 citations

TL;DR

This paper introduces the concept of almost minimizers for fractional variational problems involving the fractional Laplacian via the Caffarelli-Silvestre extension. It establishes that for a range of parameters, almost minimizers of the fractional obstacle problem and almost fractional harmonic functions exhibit almost Lipschitz or $C^{1,eta}$ regularity, extending classical regularity theory to nonlocal settings with controlled energy deviation.

ABSTRACT

In this paper we introduce a notion of almost minimizers for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli-Silvestre extension. In particular, we study almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle. We show that for a certain range of parameters, almost minimizers are almost Lipschitz or $C^{1,β}$-regular.

Motivation & Objective

• To extend the theory of minimizers to 'almost minimizers' in nonlocal variational problems governed by the fractional Laplacian.
• To study the regularity properties of functions that nearly minimize the weighted Dirichlet energy associated with the fractional Laplacian.
• To establish almost $C^{1,eta}$ and almost Lipschitz regularity for solutions to the fractional obstacle problem and fractional harmonic functions under energy deviation control.
• To provide a framework for analyzing solutions with controlled energy deviation, generalizing classical results to nonlocal settings.

Proposed method

• Define almost $s$-fractional harmonic functions via a gauge function $\omega(r)$ controlling the energy deviation from minimality in the Caffarelli-Silvestre extension setting.
• Use the Caffarelli-Silvestre extension to transform the nonlocal problem into a local degenerate elliptic problem in $\mathbb{R}^{n+1}_+$ with weight $|y|^a$.
• Employ weighted Sobolev spaces and the theory of $L_a$-harmonic functions to analyze the regularity of extensions.
• Apply orthogonal decomposition of $L_a$-harmonic homogeneous polynomials and use their properties to control growth and regularity.
• Use the maximum principle and weighted spherical averages to derive $L^\infty$ bounds from $L^2$ boundary data.
• Establish real analyticity of $L_a$-harmonic functions via series expansion in an orthonormal basis of homogeneous harmonic polynomials.

Experimental results

Research questions

• RQ1 Under what conditions do functions that nearly minimize the fractional Dirichlet energy exhibit regularity similar to minimizers?
• RQ2 Can the regularity theory for minimizers of the fractional obstacle problem be extended to functions with controlled energy deviation?
• RQ3 How does the gauge function $\omega(r)$ influence the regularity of almost minimizers in the fractional setting?
• RQ4 What is the precise regularity class (e.g., $C^{1,\beta}$) of almost fractional harmonic functions under energy deviation constraints?
• RQ5 To what extent do the regularity properties of $L_a$-harmonic functions in the extended space propagate to the original fractional solutions?

Key findings

• Almost $s$-fractional harmonic functions with a gauge function $\omega(r)$ satisfying $\omega(r) \to 0$ as $r \to 0$ are almost Lipschitz continuous in the original domain.
• For a certain range of parameters, almost minimizers of the fractional obstacle problem with zero obstacle are almost $C^{1,\beta}$ regular.
• The Caffarelli-Silvestre extension transforms the nonlocal problem into a local degenerate elliptic problem, enabling the use of classical regularity techniques.
• Solutions to the extended problem are real analytic in the interior of the domain, and this analyticity propagates to the trace on $\mathbb{R}^n$.
• The $L_a$-harmonic extension of a function in $\mathcal{L}_s(\mathbb{R}^n)$ is real analytic in $\mathbb{R}^{n+1}_\pm$ and continuous up to the boundary.
• If the trace of an $L_a$-harmonic function vanishes on $\mathbb{R}^n$, then the function itself vanishes identically in the extended space.