Some constancy results for nematic liquid crystals and harmonic maps

Chou, Kai Seng; Zhu, Xi-Ping

doi:10.1016/s0294-1449(16)30169-x

Cited by 7 publications

(5 citation statements)

References 8 publications

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“…In elliptic equations these identities are used to prove sharp nonexistence results, partial regularity of solutions, concentration phenomena, unique continuation properties, or rigidity results [34,38,13,23,52,53]. Moreover, they are also frequently used in hyperbolic equations, control theory, harmonic maps, and geometry [4,50,8,9,46,28,35].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Pohozaev identities for anisotropic integrodifferential operators

Ros‐Oton

Serra

Valdinoci

2017

Communications in Partial Differential Equations

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Abstract. We find and prove new Pohozaev identities and integration by parts type formulas for anisotropic integro-differential operators of order 2s, with s ∈ (0, 1).These identities involve local boundary terms, in which the quantity u/d s | ∂Ω plays the role that ∂u/∂ν plays in the second order case. Here, u is any solution to Lu = f (x, u) in Ω, with u = 0 in R n \ Ω, and d is the distance to ∂Ω.

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Section: Introduction and Resultsmentioning

confidence: 99%

Pohozaev identities for anisotropic integrodifferential operators

Ros‐Oton

Serra

Valdinoci

2017

Communications in Partial Differential Equations

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“…This approach has been used mostly to prove the non-existence of non-trivial solutions, see [14,24,57,59], and [55,63] for historical references. Such a method is for instance at the basis of the result of [14] on the Brezis-Nirenberg problem.…”

Section: Introductionmentioning

confidence: 99%

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Dolbeault

Stańczy

2009

Ann. Henri Poincaré

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Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

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“…These identities are used to show sharp nonexistence results, monotonicity formulas, energy estimates for ground states in R n , unique continuation properties, radial symmetry of solutions, or uniqueness results. Moreover, they are frequently used in control theory [3], wave equations [2], geometry [32,16,24], and harmonic maps [5]. For example, the controllability of the linear wave equation u tt − ∆u = 0 uses energy estimates (in terms of the boundary contribution) which follow from the Pohozaev identity.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Local integration by parts and Pohozaev identities for higher order fractional Laplacians

Ros‐Oton¹,

Serra²

2014

Preprint

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We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian (−∆) s with s > 1. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case s ∈ (0, 1).As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions (−∆) s φ = λφ in Ω, φ ≡ 0 in R n \ Ω.

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Some constancy results for nematic liquid crystals and harmonic maps

Abstract: Some constancy results for nematic liquid crystals and harmonic maps

Cited by 7 publications

References 8 publications

Pohozaev identities for anisotropic integrodifferential operators

Pohozaev identities for anisotropic integrodifferential operators

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Local integration by parts and Pohozaev identities for higher order fractional Laplacians

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