The Landis Conjecture For Nonlocal Elliptic Operators: Polynomial Decay

TL;DR

This paper proves a unique continuation property at infinity for certain nonlocal elliptic operators, showing that decay rates are polynomial rather than exponential, which advances understanding of the Landis conjecture in nonlocal contexts.

Contribution

It establishes a new polynomial decay unique continuation result for nonlinear nonlocal elliptic operators, including the fractional Laplacian, revealing the nonlocal decay behavior in Landis conjecture.

Findings

Unique continuation at infinity for nonlinear nonlocal operators.

Decay at infinity is polynomial, not exponential.

Results apply to fractional Laplacian and similar operators.

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.