Semilinear integro-differential equations, I: Odd solutions with respect to the Simons cone

Felipe-Navarro, Juan-Carlos; Sanz-Perela, Tomás

doi:10.1016/j.jfa.2019.108309

Cited by 7 publications

(27 citation statements)

References 22 publications

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“…In the fractional case L = (−∆) s , existence and uniqueness are shown in [5,3,4] by using the extension problem. For more general integro-differential operators, we can refer to the work by Cozzi and Passalacqua [8] where they prove existence, uniqueness (up to translations), and some qualitative properties of layer solutions (see [12] for further properties). Here, we prove nondegeneracy: Theorem 1.3.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Uniqueness for linear integro-differential equations in the real line and applications

Felipe-Navarro

2021

Preprint

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In this work we prove the uniqueness of solutions to the nonlocal linear equation Lϕ − c(x)ϕ = 0 in R, where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem Lu = f (u) in R when the nonlinearity is of Allen-Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli-Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Uniqueness for linear integro-differential equations in the real line and applications

Felipe-Navarro

2021

Preprint

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“…Some comments about antisymmetric functions. Maximum principles in the presence of an odd symmetry have been already treated in the field of integrodifferential equations (see for instance [38,33,19,32,30]). The main idea in this setting is that the operator can be rewritten as a different integro-differential operator acting only on functions defined in the halfspace, but still with a positive kernel (under certain assumptions on the original operator).…”

Section: −Dmentioning

confidence: 99%

The stickiness property for antisymmetric nonlocal minimal graphs

Benjamin

Dipierro

Valdinoci

2023

DCDS

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<p style='text-indent:20px;'>We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even symmetric perturbations, up to multiplicative constants).</p><p style='text-indent:20px;'>In proving this result, one also establishes an odd symmetric version of the maximum principle for nonlocal minimal graphs, according to which the odd symmetric minimizer is positive in the direction of the positive bump and negative in the direction of the negative bump.</p>

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“…In the fractional case L = (− ) s , existence and uniqueness are shown in [3][4][5] by using the extension problem. For more general integro-differential operators, we can refer to the work by Cozzi and Passalacqua [8] where they prove existence, uniqueness (up to translations), and some qualitative properties of layer solutions (see [12] for further properties). Here, we prove nondegeneracy: Theorem 1.3 Let L be an integro-differential operator of the form (1.2) satisfying the symmetry and ellipticity conditions (K1) and (K2)…”

Section: Remark 12mentioning

confidence: 99%

Uniqueness for linear integro-differential equations in the real line and applications

Felipe-Navarro

2021

Calc. Var.

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In this work we prove the uniqueness of solutions to the nonlocal linear equation $$L \varphi - c(x)\varphi = 0$$ L φ - c ( x ) φ = 0 in $$\mathbb {R}$$ R , where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem $$L u = f(u)$$ L u = f ( u ) in $$\mathbb {R}$$ R when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

show abstract

Semilinear integro-differential equations, I: Odd solutions with respect to the Simons cone

Cited by 7 publications

References 22 publications

Uniqueness for linear integro-differential equations in the real line and applications

Uniqueness for linear integro-differential equations in the real line and applications

The stickiness property for antisymmetric nonlocal minimal graphs

Uniqueness for linear integro-differential equations in the real line and applications

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