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The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay

Sebastián Flores Sepúlveda, Gabrielle Nornberg

TL;DR

The paper addresses a nonlocal analogue of the Landis conjecture for fully nonlinear integro-differential operators of order $2s$, showing that if a viscosity solution $u$ to $Iu+Vu=0$ decays faster than $o(|x|^{-(N+2s)})$ and the operator satisfies a sign condition $\lambda_1^{\pm}(I+V,G)>0$, then $u\equiv0$. The authors develop a framework based on viscosity solutions, establish a nonlocal weak Harnack inequality, and construct a positive global solution $\psi$ to $I\psi+V\psi=0$ to facilitate a comparison argument that rules out nontrivial decaying solutions. A key finding is that the Landis-type decay in the nonlocal setting is polynomial, not exponential, a result that extends beyond the fractional Laplacian to fully nonlinear operators and aligns with known polynomial decay phenomena in related nonlocal equations. This advances the understanding of unique continuation at infinity for nonlocal, nonlinear PDEs and has implications for the study of nonlocal Schrödinger-type problems.

Abstract

We obtain a unique continuation result at infinity for fully nonlinear elliptic integro-differential operators of order 2s which satisfy the maximum and minimum principles in bounded subdomains, under the decay assumption $o(|x|^{-(N+2s)})$ at infinity. Our result is new even in the case of the fractional Laplacian, as it unveils the nonlocal nature of the decay in Landis conjecture, evolving from exponential to polynomial.

The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay

TL;DR

The paper addresses a nonlocal analogue of the Landis conjecture for fully nonlinear integro-differential operators of order , showing that if a viscosity solution to decays faster than and the operator satisfies a sign condition , then . The authors develop a framework based on viscosity solutions, establish a nonlocal weak Harnack inequality, and construct a positive global solution to to facilitate a comparison argument that rules out nontrivial decaying solutions. A key finding is that the Landis-type decay in the nonlocal setting is polynomial, not exponential, a result that extends beyond the fractional Laplacian to fully nonlinear operators and aligns with known polynomial decay phenomena in related nonlocal equations. This advances the understanding of unique continuation at infinity for nonlocal, nonlinear PDEs and has implications for the study of nonlocal Schrödinger-type problems.

Abstract

We obtain a unique continuation result at infinity for fully nonlinear elliptic integro-differential operators of order 2s which satisfy the maximum and minimum principles in bounded subdomains, under the decay assumption at infinity. Our result is new even in the case of the fractional Laplacian, as it unveils the nonlocal nature of the decay in Landis conjecture, evolving from exponential to polynomial.
Paper Structure (3 sections, 12 theorems, 60 equations)

This paper contains 3 sections, 12 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Auxiliary results
  3. Main results

Key Result

Theorem 1

Let $s\in (0,1)$, $I$ be an operator as in eq:isaacs, and $V:\mathbb{R}^N \rightarrow \mathbb{R}$ be a bounded continuous function. Assume $\lambda_1^\pm ( I + V, G) >0$ for each bounded subdomain $G$ of $\mathbb{R}^N$. If $u\in L_{\omega_s}^1(\mathbb{R}^N)$ is a viscosity solution to $Iu + Vu = 0$ then $u\equiv 0$ in $\mathbb{R}^N$.

Theorems & Definitions (22)

  • Conjecture 1
  • Theorem 1
  • Remark 1
  • Definition 1: Viscosity solution
  • Lemma 2: Proposition 3.2.15 in fernandez-real_integro-differential_2024
  • Lemma 3: Pointwise evaluation, Lemma 3.3 in caffarelli_regularity_2009
  • Lemma 4: Proposition 3.2.12 in fernandez-real_integro-differential_2024
  • Proposition 5
  • Proposition 6: Weak Harnack inequality
  • proof
  • ...and 12 more