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Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory

Xavier Cabre, Iñigo U. Erneta, Juan-Carlos Felipe-Navarro

TL;DR

The article develops a general framework of null-Lagrangians and calibrations for nonlocal elliptic functionals in the presence of a field of extremals, culminating in a calibration for the energy $\\mathcal{E}_{\rm N}$ and minimality of leaves of the induced foliation. It covers a broad class of nonlocal Lagrangians, including the fractional Laplacian and fractional $p$-Dirichlet energies, and derives two key applications: (i) monotone solutions to translation-invariant nonlocal equations are minimizers within natural comparison regions, and (ii) minimizers are viscosity solutions, even without a weak comparison principle. The framework is further extended to mixed local-nonlocal energies, providing energy-comparison tools and showing how calibrations yield viscosity solutions in a broad setting. Overall, the work unifies nonlocal calibration theory with viscosity methods and offers a systematic path to handle general nonlocal variational problems lacking strong comparison principles, with explicit results available for the fractional Laplacian and related models.

Abstract

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here ``minimality'' is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions.

Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory

TL;DR

The article develops a general framework of null-Lagrangians and calibrations for nonlocal elliptic functionals in the presence of a field of extremals, culminating in a calibration for the energy and minimality of leaves of the induced foliation. It covers a broad class of nonlocal Lagrangians, including the fractional Laplacian and fractional -Dirichlet energies, and derives two key applications: (i) monotone solutions to translation-invariant nonlocal equations are minimizers within natural comparison regions, and (ii) minimizers are viscosity solutions, even without a weak comparison principle. The framework is further extended to mixed local-nonlocal energies, providing energy-comparison tools and showing how calibrations yield viscosity solutions in a broad setting. Overall, the work unifies nonlocal calibration theory with viscosity methods and offers a systematic path to handle general nonlocal variational problems lacking strong comparison principles, with explicit results available for the fractional Laplacian and related models.

Abstract

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here ``minimality'' is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions.
Paper Structure (15 sections, 13 theorems, 112 equations, 1 figure)

This paper contains 15 sections, 13 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Examples
  3. Calibrations and fields of extremals
  4. Main result
  5. An application to monotone solutions
  6. An application to the viscosity theory
  7. Outline of the article
  8. The calibration for general nonlocal functionals
  9. The calibration for functionals involving both local and nonlocal terms
  10. Application to the viscosity theory
  11. An energy comparison result for the fractional Laplacian
  12. Minimizers of functionals involving the Gagliardo seminorm are viscosity solutions
  13. General nonlocal functionals
  14. Minimality via the strong maximum principle
  15. The calibration for the nonlocal total variation

Key Result

Theorem 1.3

Let $I \subset \mathbb{R}$ be an interval and let $\Omega \subset \mathbb{R}^n$ be a bounded domain. Given a function $G_{\rm N} = G_{\rm N}(x, y, a, b)$, with $G_{\rm N}(x,y,a,b) = G_{\rm N}(y,x,b,a)$, satisfying the ellipticity condition Intro_lag:convex, let $\{u^t\}_{t \in I}$ be a field in $\ma defined in a set $\mathcal{A}_{\rm N}$ of sufficiently regular admissible functions $w \colon \math

Figures (1)

  • Figure 1: Example of a weak field for a functions $u$ in a domain $\Omega$.

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • ...and 27 more