The Liouville theorem and linear operators satisfying the maximum principle

Alibaud, Nathaël; Teso, Félix del; Endal, Jørgen; Jakobsen, Espen R.

doi:10.1016/j.matpur.2020.08.008

Cited by 13 publications

(27 citation statements)

References 19 publications

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“…As in the subelliptic setting, the derivation of gradient estimates via Bernstein-type arguments seems to be not feasible (or, at least, not straightforward), and a different procedure might be required. In this direction, we mention some recent results obtained in [3] for linear equations, which combine PDE and group theory techniques.…”

Section: The Liouville Property For Solutionsmentioning

confidence: 99%

On the Liouville property for fully nonlinear equations with superlinear first-order terms

Cirant¹,

Goffi²

2021

Preprint

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We consider in this note one-side Liouville properties for viscosity solutions of various fully nonlinear uniformly elliptic inequalities, whose prototype is, where H i has superlinear growth in the gradient variable. After a brief survey on the existing literature, we discuss the validity or the failure of the Liouville property in the model caseswhere q ≥ 0, γ > 1 and b is a suitable velocity field. Several counterexamples and open problems are thoroughly discussed.

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Section: The Liouville Property For Solutionsmentioning

confidence: 99%

On the Liouville property for fully nonlinear equations with superlinear first-order terms

Cirant¹,

Goffi²

2021

Preprint

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“…It is worth mentioning that in the papers [13,20] it is not studied the validity of a weak maximum principle as in Theorem 1.4 (i.e., the possibility of "propagating" the sign of u from R N \ Ω into Ω). On the other hand, the equivalence of (i) and (ii) is exploited in the recent paper [2] to characterize all the operators of the form (1.2) for which a Liouville-type theorem holds.…”

Section: Suppose Thatmentioning

confidence: 99%

“…The mathematical study of operators with different order is not new in itself, and indeed the literature already presents results concerning, among the others, the theory of viscosity solutions (see [32,33,3,10,19,4,5]), the Aubry-Mather theory for sums of different fractional Laplacians (see Remark 5.6 in [36]), regularization effects of Cahn-Hilliard equations (see [16]), numerics ( [11]), probability and stochastics (see [17,18,37]), symmetry results for mixed range phase transitions (see [15]), porous medium equations (see [22]), decay estimates for parabolic equations (see [26]), specific Liouville theorems for systems of equations driven by sums of fractional Laplacians (see [35,2]), fractional damping effects (see [21]), and Bernstein-type regularity results (see [14]).…”

Section: Introductionmentioning

confidence: 99%

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Biagi¹,

Dipierro²,

Valdinoci³

et al. 2020

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We develop a systematic study of the superpositions of elliptic operators with different orders, mixing classical and fractional scenarios. For concreteness, we focus on the sum of the Laplacian and the fractional Laplacian, and we provide structural results, including existence, maximum principles (both for weak and classical solutions), interior Sobolev regularity and boundary regularity of Lipschitz type.

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“…al. [2] where the authors provide a complete characterization of the translation-invariant integro-differential operators that satisfy the Liouville property in the whole space.…”

Section: Introductionmentioning

confidence: 99%

Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications

Biswas¹,

Modasiya²

2021

Preprint

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In this article we consider a class of non-degenerate elliptic operators obtained by superpositioning the Laplacian and a general nonlocal operator. We study the existence-uniqueness results for Dirichlet boundary value problems, maximum principles and generalized eigenvalue problems. As applications to these results, we obtain Faber-Krahn inequality and a one-dimensional symmetry result related to the Gibbons' conjecture. The latter results substantially extend the recent results of Biagi et. al. [7,9] who consider the operators of the form −∆ + (−∆) s with s ∈ (0, 1).

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The Liouville theorem and linear operators satisfying the maximum principle

Cited by 13 publications

References 19 publications

On the Liouville property for fully nonlinear equations with superlinear first-order terms

On the Liouville property for fully nonlinear equations with superlinear first-order terms

Mixed local and nonlocal elliptic operators: regularity and maximum principles

Mixed local-nonlocal operators: maximum principles, eigenvalue problems and their applications

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