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Novikov homology, jump loci and Massey products

Toshitake Kohno, Andrei Pajitnov (2014)

Open Mathematics

Let X be a finite CW complex, and ρ: π 1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H 1(X, ℂ) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case...

On a class of nonlinear problems involving a p ( x ) -Laplace type operator

Mihai Mihăilescu (2008)

Czechoslovak Mathematical Journal

We study the boundary value problem - d i v ( ( | u | p 1 ( x ) - 2 + | u | p 2 ( x ) - 2 ) u ) = f ( x , u ) in Ω , u = 0 on Ω , where Ω is a smooth bounded domain in N . Our attention is focused on two cases when f ( x , u ) = ± ( - λ | u | m ( x ) - 2 u + | u | q ( x ) - 2 u ) , where m ( x ) = max { p 1 ( x ) , p 2 ( x ) } for any x Ω ¯ or m ( x ) < q ( x ) < N · m ( x ) ( N - m ( x ) ) for any x Ω ¯ . In the former case we show the existence of infinitely many weak solutions for any λ > 0 . In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a 2 -symmetric version for even functionals...

On a class of ( p , q ) -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi (2017)

Communications in Mathematics

Let Ω n be a bounded starshaped domain and consider the ( p , q ) -Laplacian problem - Δ p u - Δ q u = λ ( 𝐱 ) | u | p - 2 u + μ | u | r - 2 u where μ is a positive parameter, 1 < q p < n , r p and p : = n p n - p is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the ( p , q ) -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

On a generalization of Helmholtz conditions

Radka Malíková (2009)

Acta Mathematica Universitatis Ostraviensis

Helmholtz conditions in the calculus of variations are necessary and sufficient conditions for a system of differential equations to be variational ‘as it stands’. It is known that this property geometrically means that the dynamical form representing the equations can be completed to a closed form. We study an analogous property for differential forms of degree 3, so-called Helmholtz-type forms in mechanics ( n = 1 ), and obtain a generalization of Helmholtz conditions to this case.

On a variational theory of light rays on Lorentzian manifolds

Fabio Giannoni, Antonio Masiello (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this Note, by using a generalization of the classical Fermat principle, we prove the existence and multiplicity of lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds, satisfying a suitable compactness assumption, which is weaker than the globally hyperbolicity.

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