Liouville Theorems for a General Class of Nonlocal Operators

We study existence, regularity, and positivity of solutions to linear problems involving higher-order fractional Laplacians (−∆) s for any s > 1. Using the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for s ∈ (n, n + 1) with n ∈ N odd. In contrast, we show the validity of Boggio's representation formula for all integer and fractional powers of the Laplacian s > 0. As a consequence, maximum principles hold for weak solutions in a ball. Our proofs rely on a new variational framework based on bilinear forms, on characterizations of s-harmonic functions using higher-order Martin kernels, and on a differential recurrence equation for Boggio's formula. We also discuss the case of the whole space, where maximum principles are a consequence of the fundamental solution.

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“…We now introduce the space S k s , which allows us to estimate pointwise fractional Laplacians, cf. [21,Section 2]. For s > 0 and k ∈ N let…”

Section: Properties With Respect To Smooth Functionsmentioning

confidence: 99%

On the loss of maximum principles for higher-order fractional Laplacians

Abatangelo

Jarohs

Saldaña

2018

Proc. Amer. Math. Soc.

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“…ν = 0, then the Liouville theorem is classical, and so the focus is on the nonlocal case ν = 0. For Lévy generators with a sufficiently smooth symbol, there is a Liouville theorem by Fall and Weth [5]; the required regularity of ψ increases with the dimension d ∈ N. Ros-Oton and Serra [19] established a general Liouville theorem for symmetric stable operators,…”

Section: Introductionmentioning

confidence: 99%

A Liouville theorem for Lévy generators

Kühn

2020

Positivity

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Under mild assumptions, we establish a Liouville theorem for the “Laplace” equation $$Au=0$$ A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to $$Au=0$$ A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

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“…In [63] non-existence of positive viscosity solutions was similarly addressed for Lane-Emden systems involving fractional Laplacians. In [31] a larger class of non-local operators is considered for harmonic functions, however, with a polynomial decay of its jump measure at infinity.…”

Section: Introductionmentioning

confidence: 99%

Maximum Principles and Aleksandrov--Bakelman--Pucci Type Estimates for NonLocal Schrödinger Equations with Exterior Conditions

Biswas¹,

Lőrinczi²

2019

SIAM J. Math. Anal.

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We consider Dirichlet exterior value problems related to a class of non-local Schrödinger operators, whose kinetic terms are given in terms of Bernstein functions of the Laplacian. We prove elliptic and parabolic Aleksandrov-Bakelman-Pucci type estimates, and as an application obtain existence and uniqueness of weak solutions. Next we prove a refined maximum principle in the sense of Berestycki-Nirenberg-Varadhan, and a converse. Also, we prove a weak anti-maximum principle in the sense of Clément-Peletier, valid on compact subsets of the domain, and a full anti-maximum principle by restricting to fractional Schrödinger operators. Furthermore, we show a maximum principle for narrow domains, and a refined elliptic ABP-type estimate. Finally, we obtain Liouville-type theorems for harmonic solutions and for a class of semi-linear equations. Our approach is probabilistic, making use of the properties of subordinate Brownian motion.

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Liouville Theorems for a General Class of Nonlocal Operators

Cited by 25 publications

References 13 publications

On the loss of maximum principles for higher-order fractional Laplacians

On the loss of maximum principles for higher-order fractional Laplacians

A Liouville theorem for Lévy generators

Maximum Principles and Aleksandrov--Bakelman--Pucci Type Estimates for NonLocal Schrödinger Equations with Exterior Conditions

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