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Displaying similar documents to “3D-2D asymptotic analysis for micromagnetic thin films”

Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes

Uldis Raitums (2002)

ESAIM: Control, Optimisation and Calculus of Variations

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We consider the weak closure W Z of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems where Ω 𝐑 n is a bounded Lipschitz domain, F s are strictly convex smooth functions with quadratic growth and S = { σ m e a s u r a b l e σ s ( x ) = 0 or 1 , s = 1 , , s 0 , σ 1 ( x ) + + σ s 0 ( x ) = 1 } . We show that W Z is the zero level set for an integral functional with the integrand Q being the 𝐀 -quasiconvex envelope for a certain function and the operator 𝐀 = ( curl,div ) m . If the functions F s are isotropic, then on the characteristic cone...

Some estimates for the oscillation of the deformation gradient

Vratislava Mošová (2000)

Applications of Mathematics

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As a measure of deformation we can take the difference D φ - R , where D φ is the deformation gradient of the mapping φ and R is the deformation gradient of the mapping γ , which represents some proper rigid motion. In this article, the norm D φ - R L p ( Ω ) is estimated by means of the scalar measure e ( φ ) of nonlinear strain. First, the estimates are given for a deformation φ W 1 , p ( Ω ) satisfying the condition φ | Ω = id . Then we deduce the estimate in the case that φ ( x ) is a bi-Lipschitzian deformation and φ | Ω id .

Positive solutions of the p -Laplace Emden-Fowler equation in hollow thin symmetric domains

Ryuji Kajikiya (2014)

Mathematica Bohemica

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We study the existence of positive solutions for the p -Laplace Emden-Fowler equation. Let H and G be closed subgroups of the orthogonal group O ( N ) such that H G O ( N ) . We denote the orbit of G through x N by G ( x ) , i.e., G ( x ) : = { g x : g G } . We prove that if H ( x ) G ( x ) for all x Ω ¯ and the first eigenvalue of the p -Laplacian is large enough, then no H invariant least energy solution is G invariant. Here an H invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all H invariant...

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

Mihai Mihăilescu (2008)

Czechoslovak Mathematical Journal

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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...