Liouville Type Equations with Singular Data¶and Their Applications to Periodic Multivortices¶for the Electroweak Theory

Bartolucci, Daniele; Tarantello, Gabriella

doi:10.1007/s002200200664

Cited by 190 publications

(313 citation statements)

References 35 publications

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“…We refer the reader to the books [25] and [23] for an exhaustive treatment of fluid mechanics models. Let us further rewrite Problem (3). The second equation in (3) is equivalent to rewriting the velocity field w as w = (∂ x 2 ψ, −∂ x 1 ψ).…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Two-dimensional Euler flows with concentrated vorticities

Pino

Esposito

Musso

2010

Trans. Amer. Math. Soc.

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Abstract. For a planar model of Euler flows proposed by Tur and Yanovsky (2004), we construct a family of velocity fields w ε for a fluid in a bounded region Ω, with concentrated vorticities ω ε for ε > 0 small. More precisely, given a positive integer α and a sufficiently small complex number a, we find a family of stream functions ψ ε which solve the Liouville equation with Dirac mass source,for a suitable point p = p a,ε ∈ Ω. The vorticities ω ε := −Δψ ε concentrate in the sense thatwhere the satellites a 1 , . . . , a α+1 denote the complex (α + 1)-roots of a. The point p a,ε lies close to a zero point of a vector field explicitly built upon derivatives of order ≤ α + 1 of the regular part of Green's function of the domain.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Two-dimensional Euler flows with concentrated vorticities

Pino

Esposito

Musso

2010

Trans. Amer. Math. Soc.

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“…Under the hypotheses K > 0 and α j > 0, Bartolucci and Tarantello, [5], proved a concentration-compactness result which implies that blow-up can occur only if λ belongs to the following discrete set of values Besides, they also proved an existence result for (1.9) on surfaces with positive genus and λ ∈ (8π, 16π)\Γ(α m ), generalized by Bartolucci, De Marchis and Malchiodi in [2], obtaining solvability for any λ ∈ (8π, +∞) \ Γ(α m ).…”

Section: 2)mentioning

confidence: 99%

“…It is worth to point out that the concentration compactness theorem due to Bartolucci-Tarantello [5], applied for instance in [2,3,38], is useless here because it requires K to be non negative. Indeed, we have also to rule out the possibility that some blow up occurs in S 0 ∪ S − , where we keep the notations introduced in Section 1:…”

Section: And Let K Be a Lipschitz Function On S 2 Satisfying (H1) (Hmentioning

confidence: 99%

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Existence and non existence results for the singular Nirenberg problem

Marchis

López-Soriano

2016

Calc. Var.

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ABSTRACT. In this paper we study the problem, posed by Troyanov in [47], of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a nonlinear PDE with exponential type non-linearity admitting a variational structure. In particular, we are concerned with the case where the prescribed function K changes sign. When the surface is the standard sphere, namely for the singular Nirenberg problem, we give sufficient conditions on K, concerning mainly the regularity of its nodal line and the topology of its positive nodal region, to be the Gaussian curvature of a conformal metric with assigned conical singularities.Besides, we find a class of functions on S 2 which do not verify our conditions and which can not be realized as the Gaussian curvature of any conformal metric with one conical singularity. This shows that our result is somehow sharp.

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“…(see [7,[17][18][19][20][21][22][23]). An obvious problem for the Equation (3) is the reciprocal, namely the existence of multiple blowing-up solutions with concentration points near critical points of n,m .…”

Section: Introductionmentioning

confidence: 99%